quantum instruments i extreme instruments the structure of completely positive quantum instruments is analyzed thoroughly we characterize the extreme points of the convex set of instruments the characterization holds also for nondiscrete instruments in infinite dimensions and is applied to quantum observables channels and preparations which are special cases of instruments we show that extreme instruments are discrete in finite dimensions

extremal covariant positive operator valued measures the case of a compact symmetry group given a unitary representation u of a compact group g and a transitive g space we characterize the extremal elements of the convex set of all u covariant positive operator valued measures

the norm 1 property of a quantum observable a normalized positive operator measure x e x has the norm 1 property if e x 1 whenever e x o this property reflects the fact that the measurement outcome probabilities for the values of such observables can be made arbitrarily close to one with suitable state preparations some general implications of the norm 1 property are investigated as case studies localization observables phase observables and phase space observables are considered

sample size requirements for stratified cluster randomization designs sample size requirements are provided for designs of studies in which clusters are randomized within each of several strata where cluster size itself may be a stratifying factor the approach generalizes a formula derived by woolson et al which provides sample size requirements for the cochran mantel haenszel statistic issues of data analysis are also discussed

adjustments to the mantel haenszel chi square statistic and odds ratio variance estimator when the data are clustered dependence between observations on a dichotomous variable renders invalid the usual chi square tests of independence and inflates the variances of parameter estimates such a situation occurs for example when subjects consist of members of the same family or with repeated observations on the same person in this paper we describe methods of measuring the extent of correlation or clustering we also develop methods of adjusting the mantel haenszel chi square test statistic and the variance of the mantel haenszel estimate of a common odds ratio in sets of 2 2 contingency tables

sample size requirements for estimating intraclass correlations with desired precision a method is developed to calculate the approximate number of subjects required to obtain an exact confidence interval of desired width for certain types of intraclass correlations in one way and two way anova models the sample size approximation is shown to be very accurate

linearizing non linear inverse problems and an application to inverse backscattering we propose an abstract approach to prove local uniqueness and conditional holder stability to non linear inverse problems by linearization the main condition is that in addition to the injectivity of the linearization a we need a stability estimate for a as well that condition is satisfied in particular if a a is an elliptic pseudo differential operator we apply this scheme to show uniqueness and holder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed

inverse problems for lorentzian manifolds and non linear hyperbolic equations we study two inverse problems on a globally hyperbolic lorentzian manifold m g the problems are 1 passive observations in spacetime consider observations in an open set v subset m the light observation set corresponding to a point source at q in m is the intersection of v and the light cone emanating from the point q let w subset m be an unknown open relatively compact set we show that under natural causality conditions the family of light observation sets corresponding to point sources at points q in w determine uniquely the conformal type of w 2 active measurements in spacetime we develop a new method for inverse problems for non linear hyperbolic equations that utilizes the non linearity as a tool this enables us to solve inverse problems for non linear equations for which the corresponding problems for linear equations are still unsolved to illustrate this method we solve an inverse problem for semilinear wave equations with quadratic non linearities we assume that we are given the neighborhood v of the time like path mu and the source to solution operator that maps the source supported on v to the restriction of the solution of the wave equation to v when m is 4 dimensional we show that these data determine the topological differentiable and conformal structures of the spacetime in the maximal set where waves can propagate from mu and return back to mu

a fundamental approach to the generalized eigenvalue problem for self adjoint operators the generalized eigenvalue problem for an arbitrary self adjoint operator is solved in a gelfand triple consisting of three hilbert spaces the proof is based on a measure theoretical version of the sobolev lemma and the multiplicity theory for self adjoint operators as an application necessary and sufficient conditions are mentioned such that a self adjoint operator in l2 r has generalized eigenfunctions which are tempered distributions

on scattering by time dependent potentials the 2 d case abstract we consider a real potential q x t which has compact support in space x r2 and it is damped we study the solutions of utt u q x t u 0 showing that under additional technical assumptions on q the scattering operator exists our proofs are elementary

stable recovery of time dependent coefficient from arbitrary measurements for wave equation abstract this paper concerns the stability in the inverse problem of determining the time dependent coefficient in the wave equation from partial knowledge of the dirichlet to neumann map at the boundary of a bounded domain of interest we first establish a stability estimate in the determination of the time dependent coefficient from partial boundary observations then we prove that by enlarging the set of data the unknown coefficient can be stably determined in larger regions

extreme covariant quantum observables in the case of an abelian symmetry group and a transitive value space we represent quantum observables as normalized positive operator valued measures and consider convex sets of observables which are covariant with respect to a unitary representation of a locally compact abelian symmetry group g the value space of such observables is a transitive g space we characterize the extreme points of covariant observables and also determine the covariant extreme points of the larger set of all quantum observables the results are applied to position position difference and time observables

sample size considerations for stratified cluster randomization design with binary outcomes and varying cluster size stratified cluster randomization trials crts have been frequently employed in clinical and healthcare research comparing with simple randomized crts stratified crts reduce the imbalance of baseline prognostic factors among different intervention groups due to the popularity there has been a growing interest in methodological development on sample size estimation and power analysis for stratified crts however existing work mostly assumes equal cluster size within each stratum and uses multilevel models clusters are often naturally formed with random sizes in crts with varying cluster size commonly used ad hoc approaches ignore the variability in cluster size which may underestimate overestimate the required number of clusters for each group per stratum and lead to underpowered overpowered clinical trials we propose closed form sample size formulas for estimating the required total number of subjects and for estimating the number of clusters for each group per stratum based on cochran mantel haenszel statistic for stratified cluster randomization design with binary outcomes accounting for both clustering and varying cluster size we investigate the impact of various design parameters on the relative change in the required number of clusters for each group per stratum due to varying cluster size simulation studies are conducted to evaluate the finite sample performance of the proposed sample size method a real application example of a pragmatic stratified crt of a triad of chronic kidney disease diabetes and hypertension is presented for illustration

resonances for non trapping time periodic perturbations for time periodic perturbations of the wave equation in given by a potential q t x we obtain an upper bound of the number of the resonances we establish for m n large enough a trace formula relating the iterations of the monodromy operator u mt 0 t 0 and the sum jzmj of all resonances counted with their multiplicities

stability in the determination of a time dependent coefficient for wave equations from partial data let be a c2 bounded domain of rn n 2 and fix q 0 t with t 0 we consider the stability in the inverse problem of determining a time dependent coefficient of order zero q appearing in a dirichlet initial boundary value problem for a wave equation t2u xu q t x u 0 in q from partial observations on q the observation is given by a boundary operator associated to the wave equation using suitable geometric optics solutions and carleman estimates we prove a stability estimate in the determination of q from the boundary operator

positive sesquilinear form measures and generalized eigenvalue expansions positive operator measures with values in the space of bounded operators on a hilbert space and their generalizations mainly positive sesquilinear form measures are considered with the aim of providing a framework for their generalized eigenvalue type expansions though there are formal similarities with earlier approaches to special cases of the problem the paper differs e g from standard rigged hilbert space constructions and avoids the introduction of nuclear spaces the techniques are predominantly measure theoretic and hence the hilbert spaces involved are separable the results range from a naimark type dilation result to direct integral representations and a fairly concrete generalized eigenvalue expansion for unbounded normal operators

bayesian methods of analysis for cluster randomized trials with binary outcome data we explore the potential of bayesian hierarchical modelling for the analysis of cluster randomized trials with binary outcome data and apply the methods to a trial randomized by general practice an approximate relationship is derived between the intracluster correlation coefficient icc and the between cluster variance used in a hierarchical logistic regression model by constructing an informative prior for the icc on the basis of available information we are thus able implicitly to specify an informative prior for the between cluster variance the approach also provides us with a credible interval for the icc for binary outcome data several approaches to constructing informative priors from empirical icc values are described we investigate the sensitivity of results to the prior specified and find that the estimate of intervention effect changes very little in this data set while its interval estimate is more sensitive the bayesian approach allows us to assume distributions other than normality for the random effects used to model the clustering this enables us to gain insight into the robustness of our parameter estimates to the classical normality assumption in a model with a more complex variance structure bayesian methods can provide credible intervals for a difference between two variance components in order for example to investigate whether the effect of intervention varies across clusters we compare our results with those obtained from classical estimation discuss the relative merits of the bayesian framework and conclude that the flexibility of the bayesian approach offers some substantial advantages although selection of prior distributions is not straightforward

diagonalization and representation results for nonpositive sesquilinear form measures we study decompositions of operator measures and more general sesquilinear form measures e into linear combinations of positive parts and their diagonal vector expansions the underlying philosophy is to represent e as a trace class valued measure of bounded variation on a new hilbert space related to e the choice of the auxiliary hilbert space fixes a unique decomposition with certain properties but this choice itself is not canonical we present relations to naimark type dilations and direct integrals

characterizations of the canonical phase observable in this paper we investigate various properties of phase observables which could serve to determine the canonical phase observable among the family of all phase observables we also show that any contractive weighted shift operator defines a unique phase observable and we characterize phase observables that give the most accurate phase distribution in coherent states in the classical limit

comparing completely and stratified randomized designs in cluster randomized trials when the stratifying factor is cluster size a simulation study stratified randomized designs are popular in cluster randomized trials crts because they increase the chance of the intervention groups being well balanced in terms of identified prognostic factors at baseline and may increase statistical power the objective of this paper is to assess the gains in power obtained by stratifying randomization by cluster size when cluster size is associated with an important cluster level factor which is otherwise unaccounted for in data analysis a simulation study was carried out using a crt where uk general practices were the randomized units as a template the results show that when cluster size is strongly associated with a cluster level factor which is predictive of outcome the stratified randomized design has superior power results to the completely randomized design and that the superiority is related to the number of clusters

operator integrals and phase space observables the operator integral f de of a complex valued measurable function f with respect to a positive operator measure e is considered if f is a neumark dilation of e into a projection measure then the projected operator integral pr fdf is a restriction of the operator f de necessary and sufficient conditions for the equality pr f df f de are obtained the results are applied to determine the moment operators of the phase space observables generated by the number states

estimating intraclass correlation for binary data summary this paper reviews many different estimators of intraclass correlation that have been proposed for binary data and compares them in an extensive simulation study some of the estimators are very specific while others result from general methods such as pseudo likelihood and extended quasi likelihood estimation the simulation study identifies several useful estimators one of which does not seem to have been considered previously for binary data estimators based on extended quasi likelihood are found to have a substantial bias in some circumstances

covariant localizations in the torus and the phase observables we describe all the localization observables of a quantum particle in a one dimensional box in terms of sequences of unit vectors in a hilbert space an alternative representation in terms of positive semidefinite complex matrices is furnished and the commutative localizations are singled out as a consequence we also get a vector sequence characterization of the covariant phase observables

computation of exponentials in transient data prony s method is widely used to identify the poles of a time series a numerical implementation of prony s method based on the properties of the least squares solution of the transient data matrices is described and a practical indicator of the accuracy of the computational results that is based on the singular values of the transient data matrices is derived this algorithm can be applied to time series which contain simple or multiple poles

global strichartz estimates for the wave equation with time periodic potentials abstract we obtain global strichartz estimates for the solutions u of the wave equation t 2 x v t x u f t x for time periodic potentials v t x with compact support with respect to x our analysis is based on the analytic properties of the cut off resolvent r z u t z i 1 1 where u t u t 0 is the monodromy operator and t 0 the period of v t x we show that if r z has no poles z c z 1 then for n 3 odd we have a exponential decal of local energy for n 2 even we obtain also an uniform decay of local energy assuming that r z has no poles z c z 1 and r z remains bounded for z in a small neighborhood of 0

phase space quantization and the operator moment problem we consider questions related to a quantization scheme in which a classical variable f r on a phase space is associated with a preferably unique semispectral measure ef such that the moment operators of ef are required to be of the form fk with a suitable mapping from the set of classical variables to the set of not necessarily bounded operators in the hilbert space of the quantum system in particular we investigate the situation where the map is implemented by the operator integral with respect to some fixed positive operator measure the phase space is first taken to be an abstract measurable space then a locally compact unimodular group and finally r2 where we determine explicitly the relevant operators fk for certain variables f in the case where the quantization map is implemented by a translation covariant positive operator measure in addition we consider the question under what conditions a positive operator measure is projection valued

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

a rigorous derivation of the miracle identity of three dimensional inverse scattering the large energy asymptotic behavior of scattering solutions of the three dimensional time dependent schrodinger equation is investigated the second term of the expansion leads to the miracle of newton s three dimensional inverse scattering theory

sample size for case control studies using cochran s statistic sample size determination for case control studies of chronic disease are often based on the simple 2 x 2 tabular cross classification of exposure and disease thereby ignoring stratification which may be considered in the analysis one consequence of this approach is that the sample size may be inadequate to attain a specified power and size when performing a statistical analysis on j 2 x 2 tables using cochran s 1954 biometrics 10 417 451 statistic or the mantel haenszel 1959 journal of the national cancer institute 22 719 748 statistic a sample size formula is derived from cochran s statistic and it is compared with the corresponding one derived when the data are treated as unstratified and also with two other formulas proposed for stratified data analysis the formula developed yields values slightly higher than one recently proposed by munoz and rosner 1984 biometrics 40 995 1004 which assumes that both margins of each 2 x 2 table are fixed while the present study considers only the case control margin to be fixed

gee analysis of negatively correlated binary responses a caution the method of generalized estimating equations has become almost standard for analysing longitudinal and other correlated response data however we have found that if binary responses have less than binomial variation over clusters and are modelled using exchangeable correlations prevailing software implementations may give unreliable results bounding the negative correlation away from its theoretical minimum may not always be a satisfactory solution in such instances using the independence working correlation structure and robust ses is a more trustworthy alternative

generalized coherent states and extremal positive operator valued measures we present a correspondence between positive operator valued measures povms and sets of generalized coherent states povms describe quantum observables and similar to quantum states quantum observables can also be mixed we show how the formalism of generalized coherent states leads to a useful characterization of extremal povms we prove that covariant phase space observables related to squeezed states are extremal while those related to number states are not extremal this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

on representing observables in quantum mechanics there are self adjoint operators which determine both spectral and semispectral measures these measures have very different commutativity and covariance properties this fact opens additional possibilities with regard to the physical meaning of such a self adjoint operator and its associated operator measures

gauge covariant observables and phase operators we characterize those hilbert space operators which are first moments of the gauge covariant positive operator valued measures introduced by lahti and pellonpaa we consider the suitability of such measures for describing quantum phase observing that the gauge angle defining the measure is not always closely related to the spectrum of the first moment operator we take the position that the first moment operator represents the quantal property the measure is imperfectly describing this is in contrast to the viewpoint that positive operator valued measures are the fundamental quantum observables

the leading singularity of a wave reflected by a moving boundary the propagation of singularities of waves in the presence of an impenetrable body is a well studied phenomenon on the formal level of rays snell s law states that the incident and reflected rays reflect at equal angles with the normal analytically the simplest singularity to study is the leading wavefront before multiple reflections can take place its precise nature has been derived rigorously by majda 6 and others in the case of a smooth stationary body of arbitrary shape when the body is moving the theory is less highly developed the rays no longer reflect at equal angles with the normal the reflected angle is greater if the body is locally advancing and less if the body is locally receding uniformly moving plane boundaries are studied in classical textbooks a general theory of scattering with emphasis on the resonant frequencies the scattering amplitude and the decay of the transient waves has been given by us in 3 in the present paper we calculate the wavefront which is reflected when a delta function plane wave impinges upon a three dimensional moving body our body may move or change its shape in any smooth manner as long as it remains in a fixed sphere and moves slower than the wave speed we consider only the simplest kind of wave solutions of the scalar wave equation in r4 which vanish on the space time boundary of the moving body specifically we consider the incident traveling plane wave

positive operator valued measures covariant with respect to an abelian group given a unitary representation u of an abelian group g and a subgroup h we characterize the positive operator valued measures based on the quotient group g h and covariant with respect to u

a formulation for higher dimensional inverse problems for the wave equation abstract two particular inverse scattering problems are of special interest the first concerns the discovery of a perturbation in the speed of sound by analyzing the return signal from a blast wave set off above it the second concerns the determination of a potential from the back scattering of plane waves in one dimensional problems the two cases are very closely related in higher dimensions the situation becomes much more complicated we present here a new approach to four such classical higher dimensional inverse problems for determining the coefficients of a partial differential equation including the two mentioned the idea stems from the work of deift and trubowitz 1 in one space dimension no conclusive theorem is found but the approach provides an algorithm for iteration which might lead to an existence theorem and which will be explored numerically elsewhere

normal covariant quantization maps we consider questions related to quantizing complex valued functions defined on a locally compact topological group in the case of bounded functions we generalize r werner s approach to prove the characterization of the associated normal covariant quantization maps

square integrability modulo a subgroup we prove a weak form of the frobenius reciprocity theorem for locally compact groups as a consequence we propose a definition of square integrable representation modulo a subgroup that clarifies the relations between coherent states wavelet transforms and covariant localisation observables a self contained proof of the imprimitivity theorem for covariant positive operator valued measures is given

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

phase observables phase operators and operator orderings we represent quantum phase observables as phase shift covariant normalized positive operator measures the phase operators are the first moment operators of the phase observables a phase operator determines the associated phase observable uniquely we show that the cahill glauber s ordered phase operators are determined by phase shift covariant generalized operator measures which are ordinary operator measures whenever re s 0 and phase observables when s 1 the wigner weyl quantized phase operator is not determined by any phase observable we investigate the classical limit of covariant generalized operator measures in coherent states

scattering of waves by periodically moving bodies abstract we consider solutions of the scalar wave equation vanishing on the boundary of an obstacle which undergoes periodic motion in analogy with the lax phillips theory we show that the scattering matrix as a function of frequency is holomorphic in a lower half plane and meromorphic in an upper half plane provided rays are not trapped the poles of the scattering matrix correspond to certain outgoing eigenfunctions and there is a near field expansion of finite energy solutions in terms of these eigenfunctions

scattering frequencies for the wave equation with a potential term abstract the existence of an infinite sequence of scattering frequencies for the equation u qu 0 is established where q is a real valued potential which may assume negative values this result generalizes some of the results obtained by lax and phillips in comm pure appl math 22 1969 737 787

confidence interval estimation of the intraclass correlation coefficient for binary outcome data we obtain closed form asymptotic variance formulae for three point estimators of the intraclass correlation coefficient that may be applied to binary outcome data arising in clusters of variable size our results include as special cases those that have previously appeared in the literature fleiss and cuzick 1979 applied psychological measurement 3 537 542 bloch and kraemer 1989 biometrics 45 269 287 altaye donner and klar 2001 biometrics 57 584 588 simulation results indicate that confidence intervals based on the estimator proposed by fleiss and cuzick provide coverage levels close to nominal over a wide range of parameter combinations two examples are presented

a note on infinite extreme correlation matrices abstract we give a characterization for the extreme points of the convex set of correlation matrices with a countable index set a hermitian matrix is called a correlation matrix if it is positive semidefinite with unit diagonal entries using the characterization we show that there exist extreme points of any rank

inverse scattering problem for the wave equation with time dependent potential abstract consider the equation u q t x u 0 x r 3 with a time dependent potential q t x compactly supported in x let u s c r s lloc2 r s 2 be the asymptotic wave profiles of a suitably chosen one parameter family of solutions if q1 q2 and the profiles ui related to qi i 1 2 coincide we prove that q1 q2

property c and an inverse problem for a hyperbolic equation let utt u q x t u 0 in d 0 t where d r3 is a bounded domain with a smooth boundary d t d d diam d q x t c 0 t l d suppose that for every u d f x t c1 d 0 t the value un d h s t is known where n is the outer normal to d u solves and and satisfies the initial conditions u ut 0 at t 0 then q x t is uniquely determined by the data f h f c1 d 0 t in the subset s of d 0 t consisting of the lines which make 45 with the t axis and which meet the planes t 0 and t t outside d 0 t provided that q x t is known outside s here d is the closure of d

scattering properties of wave equations with time dependent potentials we consider finite energy solutions of the perturbed wave equation with impurities which depend upon space and time u q x t u 0 with x rn t r in the case n odd 3 we prove that under suitable assumptions on the potential q x t the scattering operator exist and we analyse some of its properties finally we present a uniqueness result on an inverse scattering problem associated to the above equation in the case n 3

exponential growth for the wave equation with compact time periodic positive potential we prove the existence of smooth positive potentials v t x periodic in time and with compact support in x for which the cauchy problem for the wave equation u tt xu cv t x ud 0 has solutions with exponentially growing global and local energy moreover we show that there are resonances 2 c

on the structure of covariant phase observables we study the mathematical structure of covariant phase observables such observables can alternatively be expressed as phase matrices as sequences of unit vectors as sequences of phase states or as equivalence classes of covariant trace preserving operations covariant generalized operator measures are defined by structure matrices which form a w algebra with phase matrices as its subset the properties of the radon nikod m derivatives of phase probability measures are studied

quantum phases of a qutrit we consider various approaches to treat the phases of a qutrit although it is possible to represent qutrits in a convenient geometrical manner by resorting to a generalization of the poincare sphere we argue that the appropriate way of dealing with this problem is through phase operators associated with the algebra su 3 the rather unusual properties of these phases are caused by the small dimension of the system and are explored in detail we also examine the positive operator valued measures that can describe the qutrit phase properties

covariant phase difference observables in quantum mechanics covariant phase difference observables are determined in two different ways by a direct computation and by a group theoretical method a characterization of phase difference observables which can be expressed as the difference of two phase observables is given the classical limits of such phase difference observables are determined and the pegg barnett phase difference distribution is obtained from the phase difference representation the relation of ban s theory to the covariant phase theories is exhibited

stable determination outside a cloaking region of two time dependent coefficients in an hyperbolic equation from dirichlet to neumann map abstract in this paper we treat the inverse problem of determining two time dependent coefficients appearing in a dissipative wave equation from measured neumann boundary observations we establish in dimension n 2 stability estimates with respect to the dirichlet to neumann map of these coefficients provided that they are known outside cloaking regions moreover we prove that they can be stably recovered in larger subsets of the domain by enlarging the set of data