analytic accurate formulae for eigenenergies of a strongly anharmonic oscillator and an oscillator with a double well potential analytical formulae are presented of the first to fifth order renormalised perturbation theory rpt allowing highly accurate calculation of energy levels of a strongly anharmonic oscillator ao and an oscillator with a double well potential dwp

vectorial continued fractions and an algebraic construction of effective hamiltonians in the haromonic oscillator basis the effective hamiltonians are constructed for the polynomial interactions v r ti 1gir2i and t 4 their form factorized in terms of the vectorial continued fractions is simpler in a semirelativistic c reformulation its degeneracy in the c limit confluence of the physical and unphysical solutions is removed by solving a coupled set of the nonlinear algebraic equations in reduce

group comparisons involving missing data in clinical trials a comparison of estimates and power size for some simple approaches when using intent to treat approaches to compare outcomes between groups in clinical trials analysts face a decision regarding how to account for missing observations most model based approaches can be summarized as a process whereby the analyst makes assumptions about the distribution of the missing data in an attempt to obtain unbiased estimates that are based on functions of the observed data although pointed out by rubin as often leading to biased estimates of variances an alternative approach that continues to appear in the applied literature is to use fixed value imputation of means for missing observations the purpose of this paper is to provide illustrations of how several fixed value mean imputation schemes can be formulated in terms of general linear models that characterize the means of distributions of missing observations in terms of the means of the distributions of observed data we show that several fixed value imputation strategies will result in estimated intervention effects that correspond to maximum likelihood estimates obtained under analogous assumptions if the missing data process has been correctly characterized hypothesis tests based on variances estimated using maximum likelihood techniques asymptotically have the correct size in contrast hypothesis tests performed using the uncorrected variance obtained by applying standard complete data formula to singly imputed data can provide either conservative or anticonservative results surprisingly under several non ignorable non response scenarios maximum likelihood based analyses can yield equivalent hypothesis tests to those obtained when analysing only the observed data

application of gee procedures for sample size calculations in repeated measures experiments derivation of the minimum sample size is an important consideration in an applied research effort when the outcome is measured at a single time point sample size procedures are well known and widely applied the corresponding situation for longitudinal designs however is less well developed in this paper we adapt the generalized estimating equation gee approach of liang and zeger to sample size calculations for discrete and continuous outcome variables the non central version of the wald 2 test is considered we use the damped exponential family of correlation structures described in munoz et al for the working correlation matrix among the repeated measures we present a table of minimum sample sizes for binary outcomes and discuss extensions that account for unequal allocation staggered entry and loss to follow up 1998 john wiley sons ltd

approximate the dynamical behavior for stochastic systems by wong zakai approaching abstract random invariant manifolds and foliations play an important role in the study of the qualitative dynamical behaviors for nonlinear stochastic partial differential equations in a general way these random objects are difficult to be visualized geometrically or computed numerically the current work provides a perturbation approach to approximate these random invariant manifolds and foliations after briefly discussing the existence of random invariant manifolds and foliations for a class of stochastic systems driven by additive noises the corresponding wong zakai type of convergence result in path wise sense is established

small sample inference for fixed effects from restricted maximum likelihood restricted maximum likelihood reml is now well established as a method for estimating the parameters of the general gaussian linear model with a structured covariance matrix in particular for mixed linear models conventionally estimates of precision and inference for fixed effects are based on their asymptotic distribution which is known to be inadequate for some small sample problems in this paper we present a scaled wald statistic together with an f approximation to its sampling distribution that is shown to perform well in a range of small sample settings the statistic uses an adjusted estimator of the covariance matrix that has reduced small sample bias this approach has the advantage that it reproduces both the statistics and f distributions in those settings where the latter is exact namely for hotelling t2 type statistics and for analysis of variance f ratios the performance of the modified statistics is assessed through simulation studies of four different reml analyses and the methods are illustrated using three examples

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

a noniterative sample size procedure for tests based on t distributions a noniterative sample size procedure is proposed for a general hypothesis test based on the t distribution by modifying and extending guenther s approach for the one sample and two sample t tests the generalized procedure is employed to determine the sample size for treatment comparisons using the analysis of covariance ancova and the mixed effects model for repeated measures in randomized clinical trials the sample size is calculated by adding a few simple correction terms to the sample size from the normal approximation to account for the nonnormality of the t statistic and lower order variance terms which are functions of the covariates in the model but it does not require specifying the covariate distribution the noniterative procedure is suitable for superiority tests noninferiority tests and a special case of the tests for equivalence or bioequivalence and generally yields the exact or nearly exact sample size estimate after rounding to an integer the method for calculating the exact power of the two sample t test with unequal variance in superiority trials is extended to equivalence trials we also derive accurate power formulae for ancova and mixed effects model for repeated measures and the formula for ancova is exact for normally distributed covariates numerical examples demonstrate the accuracy of the proposed methods particularly in small samples

a simplified fixed point perturbation theory ii an application to the first few pade potentials for pt i see ibid vol 20 p 907 1987 in the radial schrodinger equation for bound states the central potentials v r vcoulombb vshort range are considered with vshort range replaced by its p j p pade approximants p j or 1 the earlier method is completed and a few coefficients of the fixed point perturbation theory asymptotic expansions of the effective hamiltonians are evaluated in reduce for p or 3

on the approximation of stochastic differential equation and on stroock varadhan s support theorem on demontre un theoreme d approximation pour l equation differentielle stochastique dx t b t x t dt i t x t o dm i t dans le cas ou m t m 1 t m l t est une semi martingale continue approchee par des semi martingales continues il sert a generaliser le theroreme du suppport de stroock varadhan au cas ou les coefficients b et i sont non bornes

a parametric family of correlation structures for the analysis of longitudinal data in epidemiological settings we are often faced with numerous short time series and a parsimonious parametrization of the correlation structure is desired in order to optimize the efficiency of the estimation procedure we propose a damped exponential correlation structure for modeling multivariate gaussian outcomes the correlation between two observations separated by s units of time is modeled as gamma s theta where gamma is the correlation between elements separated by one s unit and theta is a damping parameter for theta 0 theta 1 and theta infinity the correlation structures of compound symmetry first order autoregressive and first order moving average processes are obtained although the ar 2 dependency structure and the combination of random effects and ar 1 errors are not special cases of the proposed parametric family these structures can be well approximated within the family for short time series maximum likelihood methods for parameter estimation and interpretations of intermediate models 0 less than theta less than 1 are discussed in the context of modeling pulmonary function in an adult population in the netherlands and t cell subsets in homosexual men infected with human immunodeficiency virus type i

schrodinger equation as recurrences i band matrix hamiltonians the author describes a new algebraic treatment of the chain model eigenvalue problem it is based on the 2t 1 diagonal and asymptotically smooth structure of the corresponding hamiltonians the formalism is illustrated on the phenomenological doubly anharmonic oscillators

sample size determination for hierarchical longitudinal designs with differential attrition rates we consider the problem of sample size determination for three level mixed effects linear regression models for the analysis of clustered longitudinal data three level designs are used in many areas but in particular multicenter randomized longitudinal clinical trials in medical or health related research in this case level 1 represents measurement occasion level 2 represents subject and level 3 represents center the model we consider involves random effects of the time trends at both the subject level and the center level in the most common case we have two random effects constant and a single trend at both subject and center levels the approach presented here is general with respect to sampling proportions number of groups and attrition rates over time in addition we also develop a cost model as an aid in selecting the most parsimonious of several possible competing models i e different combinations of centers subjects within centers and measurement occasions we derive sample size requirements i e power characteristics for a test of treatment by time interaction s for designs based on either subject level or cluster level randomization the general methodology is illustrated using two characteristic examples

the anharmonic oscillator problem a new algebraic solution the second order one dimensional or radial differential schrodinger equation with the potential v r mu r2 nu r4 may be re interpreted as a difference equation of the fourth order indeed the hamiltonian is a pentadiagonal matrix in the standard harmonic oscillator basis mod n n 0 1 thus the authors construct its four independent general solutions by purely algebraic means via expansions in powers of n 1 14 next preserving the analogy between difference and differential equations the physical wavefunctions n mod psi and their energies are determined by the matching of the jost and regular solutions finally using the simplest matching conditions at the origin the resulting numerical algorithm is demonstrated to be both stable and quickly convergent

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

an impact of noise on invariant manifolds in nonlinear dynamical systems invariant manifolds provide geometric structures for understanding dynamical behavior of nonlinear systems however these nonlinear systems are often subject to random fluctuations or noises it is thus desirable to quantify the impact of noises on the invariant manifolds when the noise intensity is small this impact is estimated via asymptotic analysis in the context of liapunov perron formulation namely the random invariant manifold is represented as a perturbation of the deterministic invariant manifold with a well defined bound for the deviation

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

planning group sizes in clinical trials with a continuous outcome and repeated measures simple formulae have been derived for the accurate computation of sample sizes when the two sample t test is used required sample sizes may be reduced by taking repeated measures especially if correlations are about equal in a trial with two parallel groups repeated measures may be summarized by a pre treatment mean and a post treatment mean in a cross over trial measures may be repeated within each treatment period in case the difference between two independent groups increases in time individual slopes may be used if the correlation between two measures hardly depends on distance in time between these measures however it may be more efficient to use only the first and last measure in the statistical test if correlations strongly depend on time distance

sample size calculation for planning group sequential longitudinal trials procedures are developed in this paper for sample size calculations for planning a group sequential longitudinal trial with various correlation structures using a test statistic based on generalized estimating equations lee kim and tsiatis and group sequential boundaries based on type i error spending functions lan and demets

power calculations for general linear multivariate models including repeated measures applications abstract recently developed methods for power analysis expand the options available for study design we demonstrate how easily the methods can be applied by 1 reviewing their formulation and 2 describing their application in the preparation of a particular grant proposal the focus is a complex but ubiquitous setting repeated measures in a longitudinal study describing the development of the research proposal allows demonstrating the steps needed to conduct an effective power analysis discussion of the example also highlights issues that typically must be considered in designing a study first we discuss the motivation for using detailed power calculations focusing on multivariate methods in particular second we survey available methods for the general linear multivariate model glmm with gaussian errors and recommend those based on f approximations the treatment includes coverage of the multivariate and univariate approaches to repeated measures manova anova multivariate regression and

energy eigenvalues of quartic oscillators in d 3 dimensions the authors present a simple approximate analytical expression for the energy eigenvalues e d of the pure quartic oscillation in d or 3 dimensions the formula reproduces with high accuracy the results from accurate numerical computations reported in the literature the eigenvalue en l d for given n is seen to decrease as l increases contrary to the prediction of the quigg rosner formula for d 3 which is incorrect in this respect they also give a generalisation from the pure quartic to the quartic anharmonic oscillator of our formula good for any d this formula involves four parameters one less in the pure quartic case of which all but one are obtainable by consideration of the wkb limit the values of all the parameters are independent of the dimension d

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

power analyses for longitudinal study designs with missing data existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns although the pattern of missing data can be assessed during data analysis it is unknown during the design phase of a study the random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis in this paper we model the occurrence of missing data with a two state first order markov process and integrate the modelling information into the power function to account for random missing data patterns the markov model is easily specified to accommodate different anticipated missing data processes we develop this approach for the two most popular longitudinal models the generalized estimating equations gee and the linear mixed effects model under the missing completely at random mcar assumption for gee we also limit our consideration to the working independence correlation model the proposed methodology is illustrated with numerous examples that are motivated by real study designs

potential v r μr2 νr4 and mutual equivalence of the two recent algebraic approaches to schrodinger equations the authors consider the anharmonic oscillator hamiltonian and solve the corresponding schrodinger equation by means of the recently proposed fixed point fp perturbation theory the fp formalism resembles an introduction of the creation and annihilation operators uses a vectorial generalisation of continued fractions introduces the inverse model space dimension as a natural and controllably small expansion parameter and is shown to define a smooth exact effective truncation of the hamiltonian matrix it also reproduces the recent n 1 asymptotic formulae for the wavefunctions

sample size requirement for repeated measurements in continuous data in this paper we extend bloch s discussion on the usefulness and the limitations in the application of repeated measurements per subject in study designs we derive general sample size formulae for any finite number of comparison groups to calculate the required number of subjects with repeated measurements that do not have to be conditionally independent for fixed total cost we discuss the optimal sample allocation for repeated measurements needed to maximize the power and the underestimation when using bloch s sample size formula if in the hypothesis testing procedure the variance parameters are unknown we have also included a quantitative investigation of the effectiveness of taking repeated measurements per subjects to reduced the required number of subjects for a given power at a given alpha level

singular anharmonicities and the analytic continued fractions the potential v r 2r2 2r 4 is investigated and its bound states are constructed by a generalized hill determinant method it is shown that the binding energy e and another free parameter in the underlying laurent series representation of the wave function may be obtained from a coupled pair of hill determinant equations

an exact solution of the schrodinger equation for a multiterm potential the authors show that the schrodinger equation with the potential bx cx2 dx3 ex4 is exactly solvable on the half line x or 0 provided two simple relations between b c d and e hold some remarks concerning the b d 0 case are made

approximation and support theorems in modulus spaces we prove an approximation theorem for stochastic differential equations under rather weak smoothness conditions on the coefficients when the driving semimartingales are approximated by continuous semimartingales in probability and the solutions are considered in several banach spaces defined in terms of different types of the modulus of continuity hence stroock varadhan s support theorem is obtained in these spaces in particular in appropriate besov and holder spaces

bender wu branch points in the cubic oscillator the analytic continuation of the resonances of the cubic anharmonic oscillator to complex values of the coupling constant is studied with semiclassical and numerical methods bender wu branch points at which level crossing occurs are calculated and labelled by a process of analytic continuation the different resonances are the values that a single analytic function takes on different sheets of a riemann surface whose topology is described

harmonic oscillations in a quasi relativistic regime a square root anharmonic oscillator potential is analysed as a model which simulates a quasi relativistic squeezing of the harmonic oscillator spectrum several eligible namely perturbative variational hill determinant and riccati pade methods of construction of its bound states are compared

non hermitian matrix description of the calpcalt symmetric anharmonic oscillators the symmetric differential schrodinger equation h e with the operator h h x p2 ax4 i x3 cx2 i x h x on l2 is studied at a 0 it is rearranged as a linear algebraic diagonalization with rigorous proof our non variational construction of bound states offers an infinite dimensional analogue to the recent finite dimensional quasi exact solution available at the less common a 0

sample size calculations for two group repeated measures experiments vonesh and schork 1986 biometrics 42 601 610 presented a statistical methodology for computing the minimum sample size required for the within subjects repeated measures design they applied hotelling s t2 analysis and demonstrated the utility of these techniques under general covariance structures in this paper we extend these procedures to the between subjects repeated measures design when there are two treatment groups under consideration the multivariate analysis of variance approach to analyzing repeated measurements is considered and this model also resolves to hotelling s t2 analysis two models are put forward to contend with those situations where the correlation structure among the repeated measures is unknown these include the autoregressive model and the assumption of compound symmetry and they are compared and contrasted throughout tables of the minimum sample sizes required for several hypotheses arising from repeated measures designs are presented

fixed point perturbation theory and the potential r2 λr2 1 gr2 i analysis of convergence the simplest illustration of the recently suggested asymptotic perturbative approach to the band matrix hamiltonians is found in the harmonic oscillator complemented by the non polynomial anharmonicity lambda r2 1 gr2 in the paper the detailed construction of the effective hamiltonian is given and the convergence of its fixed point expansion is shown

one dimensional schrodinger operators with symmetric zero range potentials non hermitian hamiltonians appearing as operator realizations of symmetric symmetric in physical literature zero range singular perturbations of one dimensional schrodinger operators are studied in particular hamiltonians with a real spectrum are described in terms of parameters of singular perturbations and moreover it is shown that only part of them are similar to hermitian ones in this case they can be used as exactly solvable models of symmetric quantum mechanics

singular anharmonicities and the analytic continued fractions ii the potentials v r ar2 br 4 cr 6 the c 0 results of paper i j math phys 30 23 1989 are extended in spite of the presence of an additional coupling constant the laurent series solutions of the schrodinger equation that are obtained remain similar to mathieu functions indeed the recurrences for coefficients preserve their three term character their analytic continued fraction solutions still converge etc the formulas become even slightly simpler for c 0 due to a certain symmetry of the equations to be solved an acceleration of convergence is better understood and a few numerical illustrations of efficiency are also delivered

open perturbation and the riccati equation algebraic determination of the quartic anharmonic oscillator energies and eigenfunctions an algebraic procedure is proposed for the analytical solution of schrodinger equations that can be viewed as a factorizable equation with an additional potential v x once v x has been expanded in a series of suitable x basis functions u u x which are specific to each factorization type the solution of the riccati equation associated with the given equation is performed by means of an open perturbation technique i e at each order of the perturbation an additional balance u dependent term is introduced so that the resulting equation becomes solvable since the unperturbed potential involves the whole given potential and since the balance term is expected to be small improved results are expected at low orders of the perturbation even at the zeroth order the procedure well adapted to the use of computer algebra is applied to the solution of the gx4 anharmonic oscillator equation by means of very simple algebraic manipulations the trend of the exact values of the energies is rather well repr

on the existence of compressive solitary waves in compacting media necessary conditions for the existence of compressive solitary wave solutions of a partial differential equation derived by scott and stevenson 1986 which describes the two phase fluid flow in a medium compacting under gravity are derived it is shown that for compressive solitary wave solutions to exist which satisfy certain boundary conditions it is necessary that n m 1 where n and m are the exponents in power laws relating the permeability of the medium and the viscosity of the solid matrix respectively to the voidage the effect of the value of the exponents n and m on the shape of the solitary wave is investigated by using existing analytical solutions and new numerical solutions

strong coupling expansions for the symmetric oscillators we study the traditional problem of convergence of perturbation expansions when the hermiticity of the hamiltonian is relaxed to a weaker symmetry an elementary and quite exceptional cubic anharmonic oscillator is chosen as an illustrative example of such models we describe its perturbative features paying particular attention to the strong coupling regime efficient numerical perturbation theory proves suitable for such a purpose

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

the support of the solution to a hyperbolic spde in this paper we prove stroock varadhan type theorems for the topological support of a hyperbolic stochastic partial differential equation in the holder norm for 0 1 2 our approach is based on absolutely continuous transformations of defined using non homogeneous approximations of the brownian sheet

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

a family of complex potentials with real spectrum we consider a two parameter non hermitian quantum mechanical hamiltonian operator that is invariant under the combined effects of parity and time reversal transformations numerical investigation shows that for some values of the potential parameters the hamiltonian operator supports real eigenvalues and localized eigenfunctions in contrast with other parity times time reversal symmetric models which require special integration paths in the complex plane our model is integrable along a line parallel to the real axis

sample size calculations based on slopes and other summary statistics sample size calculations based on two sample comparisons of slopes have been reported by many this paper extends such discussions to include summary statistics other than slopes such as post baseline means change scores and final observations specifically sample size formulas for analyses based on a broad class of summary statistics are presented with modifications proposed to allow for missing data caused by staggered entry and random dropouts the formulas developed are used to illustrate how required sample size is affected by summary statistic choice variance parameters the type of treatment difference of interest and the manner in which incomplete observations are used in the analysis an example based on longitudinal data from the muscatine study is presented

repeated measures in clinical trials analysis using mean summary statistics and its implications for design this paper explores the use of simple summary statistics for analysing repeated measurements in randomized clinical trials with two treatments quite often the data for each patient may be effectively summarized by a pre treatment mean and a post treatment mean analysis of covariance is the method of choice and its superiority over analysis of post treatment means or analysis of mean changes is quantified as regards both reduced variance and avoidance of bias using a simple model for the covariance structure between time points quantitative consideration is also given to practical issues in the design of repeated measures studies the merits of having more than one pre treatment measurement are demonstrated and methods for determining sample sizes in repeated measures designs are provided several examples from clinical trials are presented and broad practical recommendations are made the examples support the value of the compound symmetry assumption as a realistic simplification in quantitative planning of repeated measures trials the analysis using summary statistics makes no such assumption however allowance in design for alternative non equal correlation structures can and should be made when necessary

sample size determination for constrained longitudinal data analysis the longitudinal data analysis model proposed by liang and zeger sankhy indian j stat ser b 2000 62 134 148 uses the baseline as well as postbaseline values as the dependent variables and the baseline mean responses are constrained to be the same across treatment groups due to randomization compared with the conventional longitudinal analysis of covariance this approach can correctly estimate the variance of within group mean changes and achieve the specified coverage probabilities general results on the sample size and power calculations for this model in the presence of missing data are obtained the sample size relationship between the constrained and unconstrained longitudinal data analysis is established simple expressions for sample size calculation are obtained for the compound symmetry and first order autoregressive correlation structures the sensitivity of the sample size requirement to the configuration of correlation structure and retention pattern is assessed the performance of several ad hoc approximations for longitudinal data analysis sample size calculation is evaluated simulation studies are conducted to assess the validity of the proposed sample size formulas with deviation from normality the sample size formulas are also illustrated in detail using real clinical trial data copyright 2008 john wiley sons ltd

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

schrodinger equation as recurrences iii partitioning approach for pt ii see ibid vol 17 no 8 p 1603 10 1984 in the generalised lanczos basis the author partitions the hamiltonian to tridiagonal form and mutatis mutandis apply the ideas of i and ii it solves the schrodinger eigenvalue problem again the resulting method modifies and complements the formalism and simplifies its use and interpretation as an example of application the unique asymptotical effective hamiltonian is derived also for the anharmonic oscillators the convergence of its continued fractional definition is proved and the compact formula for the higher order asymptotic corrections is found

closed form reml estimators and sample size determination for mixed effects models for repeated measures under monotone missingness we derive the closed form restricted maximum likelihood estimator and kenward roger s variance estimator for fixed effects in the mixed effects model for repeated measures mmrm when the missing data pattern is monotone as an important application of the analytic result we present the formula for calculating the power of treatment comparison using the wald t test with the kenward roger adjusted variance estimate in mmrm it allows adjustment for baseline covariates without the need to specify the covariate distribution in randomized trials a simple two step procedure is proposed to determine the sample size needed to achieve the targeted power the proposed method performs well for both normal and moderately non normal data even in small samples n 20 in simulations an antidepressant trial is analyzed for illustrative purposes copyright 2017 john wiley sons ltd

splitting in a double minimum potential with almost twofold degenerate lower levels an analytic continuation procedure using taylor series is utilised to produce very accurate wavefunctions and eigenvalues for a double minimum potential having almost degenerate eigenvalues comparison is made with results obtained by other means

a wong zakai approximation for random invariant manifolds random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences but these random objects are difficult to be visualized geometrically or computed numerically the current work provides a perturbation approach to approximate these random invariant manifolds we first discuss the existence of a random invariant manifold for a class of stochastic evolutionary equations then we approximate the random invariant manifold by the invariant manifold of a new system with smooth colored noise i e integrated ornstein uhlenbeck processes the convergence in a pathwise wong zakai sense is shown

potential r2 λr2 1 gr2 and the analytic continued fractions the author constructs the exact wavefunctions and an analytic continued fractional green function for the one and three dimensional schrodinger equation with the potential v r h r2 r2 lambda r2 1 gr2 h 1 4 lambda g 0 in the numerical application of the formulae the energy and norm of the boundary state psi may be approximated by both their lower and upper estimates

on the simultaneous eigenproblem for the x2 λ x2 1 gx2 1 interaction extension of gallas results an ansatz for the eigenfunctions of the x2 x2 1 gx2 1 interaction has been recently introduced by blecher and leach using this ansatz gallas has obtained five solution pairs for 2g 3g 2 gallas results are generalised and extended here

on the power series construction of bound states i the energies as zeros of the infinite hill determinants for a broad class of potentials we show that the nonvariational so called hill determinant intuitive identification of binding energies with zeros of certain infinite determinants may be given a rigorous mathematical foundation the essence of the construction lies in an appropriate restriction of the class of the admissible ansatze this eliminates the undesirable confluence of the physical and unphysical components of r that may take place in the general case

a simple sample size formula for analysis of covariance in cluster randomized trials for cluster randomized trials with a continuous outcome the sample size is often calculated as if an analysis of the outcomes at the end of the treatment period follow up scores would be performed however often a baseline measurement of the outcome is available or feasible to obtain an analysis of covariance ancova using both the baseline and follow up score of the outcome will then have more power we calculate the efficiency of an ancova analysis using the baseline scores compared with an analysis on follow up scores only the sample size for such an ancova analysis is a factor r 2 smaller where r is the correlation of the cluster means between baseline and follow up this correlation can be expressed in clinically interpretable parameters the correlation between baseline and follow up of subjects subject autocorrelation and that of clusters cluster autocorrelation because of this subject matter knowledge can be used to provide range of plausible values for these correlations when estimates from previous studies are lacking depending on how large the subject and cluster autocorrelations are analysis of covariance can substantially reduce the number of clusters needed copyright c 2012 john wiley sons ltd

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

on efficiency of constrained longitudinal data analysis versus longitudinal analysis of covariance in randomized clinical trials measurements are often collected on each subject at a baseline visit and several post randomization time points the longitudinal analysis of covariance in which the postbaseline values form the response vector and the baseline value is treated as a covariate can be used to evaluate the treatment differences at the postbaseline time points liang and zeger 2000 sankhy the indian journal of statistics series b 62 134 148 propose a constrained longitudinal data analysis in which the baseline value is included in the response vector together with the postbaseline values and a constraint of a common baseline mean across treatment groups is imposed on the model as a result of randomization if the baseline value is subject to missingness the constrained longitudinal data analysis is shown to be more efficient for estimating the treatment differences at postbaseline time points than the longitudinal analysis of covariance the efficiency gain increases with the number of subjects missing baseline and the number of subjects missing all postbaseline values and for the pre post design decreases with the absolute correlation between baseline and postbaseline values

power analyses for longitudinal trials and other clustered designs summary existing methods for power and sample size estimation for longitudinal and other clustered study designs have limited applications in this paper we review and extend existing approaches to improve these limitations in particular we focus on power analysis for the two most popular approaches for clustered

sample sizes in the multivariate analysis of repeated measurements determination of sample sizes for comparing two or more treatments in repeated measurements experiments is considered multivariate normality of the individual s vector of repeated measures is assumed particular emphasis is placed on applications wherein the error variance covariance matrix is arbitrary positive definite sample size determination is based on power considerations associated with hotelling s t2 test and the desire to detect a specified difference between any pair of treatment means tabulated sample sizes are given in the case of an equal variance unequal covariance structure the utility of these sample sizes is also demonstrated for a more general variance covariance structure applications of these sample sizes are illustrated with two examples

the quartic oscillator the quantum quartic oscillator is treated on the basis of a singularity analytic approach to the central two point connection problem of the triconfluent case of heun s differential equation we split off the asymptotic factors by means of a specific linear transformation of the independent variable and represent the solution in terms of a jaffe expansion the result is a fourth order linear difference equation of poincare perron type the asymptotic behavior of which is significant for the connection problem this is investigated by means of the birkhoff set of the difference equation and leads to the exact eigenvalue condition of the problem

pade oscillators and a new formulation of perturbation theory being guided by the problem of bound states in potentials close to their pade approximants a new rayleigh schrodinger type perturbation theory is developed the unperturbed system is understood here in a broader sense its solutions are not needed but merely the related nondiagonal unperturbed propagator r in particular all the chain models h0 es0 h0 s0 band matrices with arbitrary perturbations are then perturbatively solvable with r constructed in terms of auxiliary matrix continued fraction fn alternatively a generalized unperturbed spectrum fn may be required as an input the algebraically constructed asymptotics of the fn s play this role in our pade examples due to s i the sturmians may also be constructed in the test evaluations of the binding energies and or couplings the simultaneous upper and lower bounds of high precision are shown to be numerically obtainable

chain models and their effective hamiltonians we develop a new formalism suitable for diagonalization of hamiltonians with 2t 1 nonzero diagonals chain models in a systematic perturbation like way we get the expansions of green and or wave functions they are generated by solution of a sequence of fixed point quadratic equations for t t dimensional matrices for t 1 and t 2 this solution is feasible by elementary means so that the respective tridiagonal and pentadiagonal chain models may be considered exactly solvable in this context in an alternative first order variational perturbation formulation the method may provide the simultaneous upper and lower energy bounds for any t

the power to detect differences in average rates of change in longitudinal studies with considerable current interest in longitudinal epidemiologic studies little is available regarding sample size requirements this paper considers a method for analysis of longitudinal data where one compares the mean rates of change for two or more groups and proposes a statistic for use in determining sample size requirements one calculates individual rates of change with least squares estimates of slopes of individuals responses regressed over time the assumption of linear change over time while clearly not applicable for some data applies to many biological measurements either as recorded or with some transformation the variances of these estimated slopes have two components within individual variability based on measurement error and length and frequency of follow up and true between individual slope variability it is assumed that measurement error is the same for all subjects so that the total variances differ due to differences in follow up the question addressed is when can one use the usual anova f statistic to compare group means of estimated slopes expected mean squares demonstrate that this f is appropriate when either each group has the same number of subjects or when each subject has the same length and frequency of follow up a procedure for computing power and sample size is presented where one can specify the maximum detectable difference in any two average slopes moment estimation and maximum likelihood estimation of variance components from prior data are discussed

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

a simplified fixed point perturbation theory i the method for a simple potential the author proposes a conversion of the standard differential radial schrodinger equation into an infinite dimensional matrix equation a phi 0 and describes its systematic non numerical solution the results may be characterised as infinite asymptotic expansions of the wavefunctions phi and or of the secular determinant an inverse green function det a the inverse model space dimension plays a role of the small perturbation parameter

sample size calculations for studies with correlated observations correlated data occur frequently in biomedical research examples include longitudinal studies family studies and ophthalmologic studies in this paper we present a method to compute sample sizes and statistical powers for studies involving correlated observations this is a multivariate extension of the work by self and mauritsen 1988 biometrics 44 79 86 who derived a sample size and power formula for generalized linear models based on the score statistic for correlated data we appeal to a statistic based on the generalized estimating equation method liang and zeger 1986 biometrika 73 13 22 we highlight the additional assumptions needed to deal with correlated data some special cases that are commonly seen in practice are discussed followed by simulation studies

a new method in the many body problem ii the earlier proposed method for exact determination of the energy spectra of hamiltonians h and the quantities where w is a finite dimensional perturbation is extended to treat a still larger class of hamiltonians in particular some two and three dimensional problems became amenable to this approach in addition the expressions for the density and local densities of states are deduced as an example an impurity problem is considered in a simple cubic lattice with short range and in a linear chain with long range interactions

approximation and support theorem in holder norm for parabolic stochastic partial differential equations the solution u t x of a parabolic stochastic partial differential equation is a random element of the space e of holder continuous functions on 0 t x 0 1 of order 1 4 e in the time variable and 1 2 e in the space variable for any e 0 we prove a support theorem in e for the law of u the proof is based on an approximation procedure in holder norm which should have its own interest using a space time polygonal interpolation for the brownian sheet driving the spde and a sequence of absolutely continuous transformations of the wiener space

schrodinger equation as recurrences ii general solutions and their physical asymptotics for pt i see ibid vol 16 no 17 p 4001 11 1983 the general schrodinger equation is converted into equivalent 2t 1 term recurrences by the generalised lanczos construction of the basis then the exact bound states are given by the t parametric determinantal formula finally the values of the free parameters are shown to follow from the hill type normalisability requirement

shape invariant potentials with pt symmetry suitable complexification of the well known hyperbolic rosen morse oscillator in one dimension is shown to give the fourth and last exactly solvable model which combines the shape and invariance

sample size estimation for gee method for comparing slopes in repeated measurements data sample size calculation is an important component at the design stage of clinical trials controlled clinical trials often use a repeated measurement design in which individuals are randomly assigned to treatment groups and followed up for measurements at intervals across a treatment period of fixed duration in studies with repeated measurements one of the popular primary interests is the comparison of the rates of change in a response variable between groups statistical models for calculating sample sizes for repeated measurement designs often fail to take into account the impact of missing data correctly in this paper we propose to use the generalized estimating equation gee method in comparing the rates of change in repeated measurements and introduce closed form formulae for sample size and power that can be calculated using a scientific calculator since the sample size formula is based on asymptotic theory we investigate the performance of the estimated sample size in practical settings through simulations

asymmetric anharmonic oscillators in the hill determinant picture a new version of the hill determinant construction of bound states is proposed it is based on the matching of two suitable power series ansatze for the wave function for the asymmetric well potential v x ax bx2 cx3 dx4 the convergence of the approximants to the physical solutions is proved for b bminimal the taylor coefficients for this anharmonic oscillator s wave function are specified by four term recurrences

sample size and power determination for clustered repeated measurements it is common in epidemiological and clinical studies that each subject has repeated measurements on a single common variable while the subjects are also clustered to compute sample size or power of a test we have to consider two types of correlation correlation among repeated measurements within the same subject and correlation among subjects in the same cluster we develop based on generalized estimating equations procedures for computing sample size and power with clustered repeated measurements explicit formulae are derived for comparing two means two slopes and two proportions under several simple correlation structures

avoided crossings of the quartic oscillator the phenomenon of avoided crossings of energy levels in the spectrum of quantum systems is well known however being of an exponentially small order it is hard to calculate in particular this is the case when the potential is generating a schrodinger equation of a type which is beyond the hypergeometric one recently there have been attempts to understand this phenomenon in connection with heun type differential equations the most famous example of this class is the quantum quartic oscillator which is governed by the triconfluent case of heun s differential equation in the following we consider situations where the fourth order potential has two minima and we calculate the avoided crossings of its eigenvalue curves in dependence on the asymmetry and the barrier height between the two wells the results are compared with those obtained from an asymptotic approach of the problem for large values of the control parameter that governs the barrier height

eigenvalues of an anharmonic oscillator ii an expression is derived for the sixth nonzero term in the wkbj approximation the six term wkbj approximation is applied to calculate the eigenvalues for the potential v x 1 2 kx2 ax4 k 0 and a 0 at low values of ah k3 1 2 the calculated results are in excellent agreement to 15 significant figures with those of banerjee et al for all quantum numbers at medium and high values of the calculated results are poor at low quantum numbers but improve rapidly as n increases a 15 significant figure accuracy is achieved at n 10 for 0 05 and at about n 15 for 20 000 for 0 5 eigenvalues are calculated to 20 significant figures and an argument is presented to show that by n 50 the calculated value is correct to 19 significant figures the accuracy further improves at higher quantum numbers

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

on exact solutions of the schrodinger equation in the one and three dimensional schrodinger equation a broad class of the regular potentials rational functions of the rational powers of r may admit exact bound state solutions in the generalised harmonic oscillator elementary form psi r rsigma polynomial exp polynomial the necessary and sufficient conditions of this phenomenon are derived in the form of coupled algebraic equations the methods of their solution and a few examples are discussed in particular the well known coulombic and oscillator solvability and the similar recent results of singh et al 1978 and whitehead et al 1982 are reproduced as the two simplest special cases