diagnostics for the exponential normal growth curve model the exponential normal regression model provides a flexible parametric approach to constructing growth charts or centile curves we develop diagnostics for this model with a special emphasis towards assessing the impact of individual cases on estimated centile curves we also propose a standardized score residual that assesses the fit of individual cases to the location scale and skewness functions respectively an application of the proposed diagnostics is presented

elementary exponential error estimates for the adiabatic approximation we present an elementary proof that the quantum adiabatic approximation is correct up to exponentially small errors for hamiltonians that depend analytically on the time variable our proof uses optimal truncation of a straightforward asymptotic expansion we estimate the terms of the expansion with standard cauchy estimates

general adiabatic evolution with a gap condition we consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity away from the rest of the spectrum the restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable

spectral theory for unitary operators on a quaternionic hilbert space the spectral theorem for a unitary operator on a quaternionic hilbert space is established and a number of related results are proved

exponentially small adiabatic invariant for the schrodinger equation we study an adiabatic invariant for the time dependent schrfdinger equation which gives the transition probability across a gap from time t to time t when the hamiltonian depends analytically on time and t oo t oo we give sufficient conditions so that this adiabatic invariant tends to zero exponentially fast in the adiabatic limit

a system of matrix equations with five variables in this paper we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations a 1 x c 1 y b 1 d 1 a 2 w c 2 z b 2 d 2 a 3 v c 3 v b 3 c 4 a 4 v b 4 c 5 a 5 x y b 5 c 6 w z d 6 e 6 v f 6 g 6 and constitute an expression of the general solution to the system when it is solvable the outcomes of this paper encompass some recognized results in the collected works in addition we establish an algorithm and a numerical example to illustrate the theory constructed in the paper

quaternionic differential operators motivated by a quaternionic formulation of quantum mechanics we discuss quaternionic and complex linear differential equations we touch only a few aspects of the mathematical theory namely the resolution of the second order differential equations with constant coefficients we overcome the problems coming out from the loss of the fundamental theorem of the algebra for quaternions and propose a practical method to solve quaternionic and complex linear second order differential equations with constant coefficients the resolution of the complex linear schrodinger equation in the presence of quaternionic potentials represents an interesting application of the mathematical material discussed in this paper

superadiabatic tracking of quantum evolution the adiabatic quantum evolution of a two state system without energy level crossings is an example of the stokes phenomenon in the latter a small subdominant exponential is an asymptotic expansion appears when a stokes line is crossed truncating the dominant asymptotic series at its least term causes the multiplier of the subdominant term to rise in a smooth compact and universal manner across the stokes line in quantum evolution this corresponds to a smooth transition universal in form between superadiabatic basis states high order wkb approximate solutions of the time dependent schrodinger equation the authors give a numerical demonstration of this previously predicted universality by constructing for two hamiltonians the superadiabatic quantum bases asymptotic to the actual evolving state universality when a stokes line is crossed is seen in the changing probability that the system makes a transition away from the superadiabatic state and occurs at that order of superadiabatic approximation corresponding to truncating the asymptotic series at its least term

controlling for a confounder monotonically related to exposure by means of isotonic quantile regression we present a new approach to evaluating the effect of a continuous exposure factor when that factor increases with a covariate for example age which is regarded as a potential confounder the basic idea is to estimate as functions of the covariate some selected quantiles of the exposure distribution under the assumption that the dependence of each quantile on the covariate is monotonic the resulting estimates are then used to divide the data into different exposure categories this method of categorizing the data implies that the covariate distribution will be almost the same in each exposure group we illustrate the approach with a study of blood pressure and cardiovascular mortality

diagonalization of bounded linear operators on separable quaternionic hilbert space by using the representation of its complex conjugate pairs we have investigated the diagonalization of a bounded linear operator on separable infinite dimensional right quaternionic hilbert space the sufficient condition for diagonalizability of quaternionic operators is derived the result is applied to anti hermitian operators which is essential for solving schrodinger equation in quaternionic quantum mechanics

slowly changing potential problems in quantum mechanics adiabatic theorems ergodic theorems and scattering we employ the recently developed multi time scale averaging method to study the large time behavior of slowly changing in time hamiltonians we treat some known cases in a new way such as the zener problem and we give another proof of the adiabatic theorem in the gapless case we prove a new uniform ergodic theorem for slowly changing unitary operators this theorem is then used to derive the adiabatic theorem do the scattering theory for such hamiltonians and prove some classical propagation estimates and asymptotic completeness

time development of exponentially small non adiabatic transitions optimal truncations of asymptotic expansions are known to yield approximations to adiabatic quantum evolutions that are accurate up to exponentially small errors in this paper we rigorously determine the leading order non adiabatic corrections to these approximations for a particular family of two level analytic hamiltonian functions our results capture the time development of the exponentially small transition that takes place between optimal states by means of a particular switching function our results confirm the physics predictions of sir michael berry in the sense that the switching function for this family of hamiltonians has the form that he argues is universal

quaternions and special relativity we reformulate special relativity by a quaternionic algebra on reals using real linear quaternions we show that previous difficulties concerning the appropriate transformations on the 3 1 space time may be overcome this implies that a complexified quaternionic version of special relativity is a choice and not a necessity

age related standards for ordinal data modelling the changes in visual acuity from 2 to 9 years of age the visual acuities of 1140 children between the ages of 2 and 9 years were measured using the sonksen silver acuity system ssas from a distance of 3 metres ordered logistic regression was used to construct smoothly changing age related reference ranges for the proportions of children at different ages expected to have acuities in each of three ordinal categories exponential functions of age fitted the data better than polynomials the resulting reference ranges can be used to derive the ages at which a given percentage of children achieve each of the acuity cut offs however since it is unlikely that all children will be available for testing at the exact appropriate age two possible alternatives are introduced and discussed first a new ordinal category that would result in a chosen outcome at a pre selected age is calculated secondly because it may be more feasible to use the current ssas system whilst varying the distance of measurement age related standards showing the distances required to give constant pass rates with the usual letter sizes are also presented

superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum during the last years several theoretical and experimental works have been devoted to different aspects of the adiabatic regime of quantum dynamics we consider here two aspects of this physically important regime the construction of approximate evolutions of the schriidinger equation called superadiabatic evolutions and the explicit computation of the probability of a transition between two nondegenerate energy levels isolated in the spectrum of the hamiltonian from the mathematical point of view the adiabatic limit for time dependent systems is of the same kind as the semiclassical limit for stationary systems it is a singular limit which cannot be tackled by usual perturbation methods recently a mathematical theory of the two considered aspects has been developed when the time dependent hamiltonian is analytic in time the transition probability for two level systems has been studied in refs l 3 and was shown to be exponentially small in the adiabaticity parameter for general systems in ref 4 on the other hand the existence of superadiabatic evolutions has been shown in ref 5 however no general formula for the transition probability between isolated levels in the adiabatic limit can be found in the literature it is the goal of this paper to establish such a formula for general unbounded hamiltonians the paper is divided into two parts we first construct a superadiabatic evolution which approximates the true evolution of the schrijdinger equation up to exponentially small correction terms with respect to the parameter controlling the adiabatic regime once the superadiabatic evolution is defined we show that the computation of the transition probability of the initial problem reduces to an effective two dimensional problem which we analyze as in ref 1 the concept of superadiabatic evolution can be traced back up to a paper of garrido 6 who considered evolutions different from the usual adiabatic evolution which he used to obtain higher order corrections to the adiabatic theorem of quantum mechanics but it is berry7v8 who emphasized the importance of such evolutions in the present problem and introduced the

analytic plane wave solutions for the quaternionic potential step by using the recent mathematical tools developed in quaternionic differential operator theory we solve the schrodinger equation in the presence of a quaternionic step potential the analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart a brief discussion on reflected and transmitted times performed by using the stationary phase method and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented the analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem up to now solved by numerical method

constructing time specific reference ranges reference ranges which take time such as age into account are often required in medicine but simple systematic and efficient statistical methods for constructing them are lacking a method is described which is based on low order polynomial curves linear quadratic or occasionally cubic together with guidelines for when and how to apply a logarithmic transformation to the variable analysed testing for departures from normality and assessment of the adequacy of the reference range which is constructed from the regression line plus or minus a multiple of the standard deviation standard statistical packages may be used to carry out the calculations the question of comparing two or more groups of patients is addressed three examples are discussed in detail

worm plot a simple diagnostic device for modelling growth reference curves the worm plot visualizes differences between two distributions conditional on the values of a covariate though the worm plot is a general diagnostic tool for the analysis of residuals this paper focuses on an application in constructing growth reference curves where the covariate of interest is age the lms model of cole and green is used to construct reference curves in the fourth dutch growth study 1997 if the model fits the measurements in the reference sample follow a standard normal distribution on all ages after a suitably chosen box cox transformation the coefficients of this transformation are modelled as smooth age dependent parameter curves for the median variation and skewness respectively the major modelling task is to choose the appropriate amount of smoothness of each parameter curve the worm plot assesses the age conditional normality of the transformed data under a variety of lms models the fit of each parameter curve is closely related to particular features in the worm plot namely its offset slope and curvature application of the worm plot to the dutch growth data resulted in satisfactory reference curves for a variety of anthropometric measures it was found that the lms method generally models the age conditional mean and skewness better than the age related deviation and kurtosis copyright 2001 john wiley sons ltd

constrained two sided coupled sylvester type quaternion matrix equations abstract in this paper we consider the solvability of a system of constrained two sided coupled generalized sylvester quaternion matrix equations some necessary and sufficient conditions for the existence of solutions to this system are derived in terms of the ranks and the generalized inverses of the coefficient matrices moreover we find an expression of the general solution to this system when the solvability conditions are satisfied finally a numerical example is presented to illustrate our results

histories of adiabatic quantum transitions the way in which the transition amplitude to an initially unoccupied state increases to its exponentially small final value is studied in detail in the adiabatic approximation for a 2 state quantum system by transforming to a series of superadiabatic bases clinging ever closer to the exact evolving state it is shown that transition histories renormalize onto a universal one in which the amplitude grows to its final value as an error function rather than via large oscillations as in the ordinary adiabatic basis the time for the universal transition is of order surd hslash delta where delta is the small adiabatic slowness parameter in perturbation theory the pre exponential factor of the final amplitude renormalizes superadiabatically from the incorrect value textstyle frac 1 3 pi for the ordinary adiabatic basis to the correct value unity the various histories could be observed in spin experiments

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

octonionic representations of gl 8 r and gl 4 c octonionic algebra being nonassociative is difficult to manipulate we introduce left right octonionic barred operators which enable us to reproduce the associative gl 8 r group extracting the basis of gl 4 c we establish an interesting connection between the structure of left right octonionic barred operators and generic 4 4 complex matrices as an application we give an octonionic representation of the four dimensional clifford algebra

precise exponential estimates in adiabatic theory general adiabatic evolutions associated to hamiltonians which admit a holomorphic extension with respect to the time variable in a complex strip and whose spectrum satisfies a gap condition are studied an explicit rate of exponential decay is given which is related to simple geometric quantities associated to the spectrum of the hamiltonian for the transition probability between the two parts of the spectrum when the evolution is taken from to

linear adiabatic theory exponential estimates a general adiabatic expansion is written down the basic result is that under appropriate smoothness conditions the adiabatic estimations can be pushed up to exponential order the results imply exponential estimates not only for hamiltonians analytic in a neighbourhood of the real axis but also for hamiltonians which in an appropriate sense beong to gevrey classes

maximum likelihood estimation of reference centiles we propose the use of centile estimates which are based on the fitting of appropriate densities by maximum likelihood in the case of cross sectional centile estimation we show that this approach will generally lead to more precise estimates than would result from the use of non parametric centile estimates when longitudinal data are available or a series of cross sectional data at different time points the maximum likelihood approach can be used to simultaneously fit densities to each cross section subject to constraints for example smoothness constraints on the parameters the variances of these centile estimates are readily obtained and missing values and unequally spaced records are easily accommodated we illustrate the procedure by means of an application using the johnson family of densities to a study of weight gain in pregnancy

quaternionic representations of magnetic groups we study magnetic groups which contain a time inversion operator besides spatial symmetries in the framework of quaternionic group representation theory we obtain a classification of these groups depending on the reducibility of their spatial part and then we cross it with the generalized frobenius schur classification that has been obtained by ourselves elsewhere ten distinct cases arise but only five of them apply to factorizable groups i e groups in which the time inversion operator appears alone and not multiplied by a spatial symmetry we supply examples for these five cases and determine which of them apply to bosonic or fermionic systems respectively finally we discuss the degeneracy of energy levels in the presence of a time reversal symmetry kramers degeneracy in quaternionic quantum mechanics obtaining right values in all cases

finite lorentz transformations automorphisms and division algebras an explicit algebraic description of finite lorentz transformations of vectors in ten dimensional minkowski space is given by means of a parametrization in terms of the octonions the possible utility of these results for superstring theory is mentioned along the way automorphisms of the two highest dimensional normed division algebras namely the quaternions and the octonions are described in terms of conjugation maps similar techniques are used to define so 3 and so 7 via conjugation so 4 via symmetric multiplication and so 8 via both symmetric multiplication and one sided multiplication the noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions

a two part model for reference curve estimation subject to a limit of detection reference curves are commonly used to identify individuals with extreme values of clinically relevant variables or stages of progression which depend naturally on age or maturation estimation of reference curves can be complicated by a technical limit of detection lod that censors the measurement from the left as is the case in our study of reproductive hormone levels in boys around the time of the onset of puberty we discuss issues with common approaches to the lod problem in the context of our pubertal hormone study and propose a two part model that addresses those issues one part of the proposed model specifies the probability of a measurement exceeding the lod as a function of age the other part of the model specifies the conditional distribution of a measurement given that it exceeds the lod again as a function of age information from the two parts can be combined to estimate the identifiable portion i e above the lod of a reference curve and to calculate the relative standing of a given measurement above the lod unlike some common approaches to lod problems the two part model is free of untestable assumptions involving unobservable quantities flexible for modeling the observable data and easy to implement with existing software the method is illustrated with hormone data from the third national health and nutrition examination survey

conditional growth charts growth charts are often more informative when they are customized per subject taking into account prior measurements and possibly other covariates of the subject we study a global semiparametric quantile regression model that has the ability to estimate conditional quantiles without the usual distributional assumptions the model can be estimated from longitudinal reference data with irregular measurement times and with some level of robustness against outliers and it is also flexible for including covariate information we propose a rank score test for large sample inference on covariates and develop a new model assessment tool for longitudinal growth data our research indicates that the global model has the potential to be a very useful tool in conditional growth chart analysis

a rejoinder on quaternionic projective representations in a series of papers published in this journal a discussion was started on the significance of a new definition of projective representations in quaternionic hilbert spaces in the present paper we give what we believe is a resolution of the semantic differences that had apparently tended to obscure the issues