fake lie groups with maximal tori iv this is a continuation of n s 1 2 3 which wc refer to as i ii and iii in this series of papers fake lie groups and adic fake lie groups are studied we will recall some of the basic definitions further notations and terminology may be found in i ii and li1 an adic fake lie group of type g g a compact connected lie group is a finite loop space x such that the classifying space is in the adic genus of bg i e bg and bx are equivalent after localisation at or any p completion x is a fake lie group of type g if x is in the genus of bg i e x and bg are p local equivalent for every prime genusoa bg denotes the adic genus and genus bg the genus of bg a maximal torus for a adic fake lie group x of type g is a map j r btx bx tx a torus such that the homotopy fiber o f f r has the homotopy type of a finite cw complex and such that rank tx rank x rank g the weyl group wx is defined to be the set of homotopy classes of homotopy equivalences q btx btx such that ft 7 q 7 in i 3 7 3 6 it is proved that wx wg as abstract groups one of the main goals was and is to determine all fake lie groups allowing a maximal torus in ii and iii this is done for simply connected lie groups and unitary groups here we will extend these results to the general case

performance bounds for estimating vector systems we propose a unified framework for the analysis of estimators of geometrical vector quantities and vector systems through a collection of performance measures unlike standard performance indicators these measures have intuitive geometrical and physical interpretations are independent of the coordinate reference frame and are applicable to arbitrary parameterizations of the unknown vector or system of vectors for each measure we derive both finite sample and asymptotic lower bounds that hold for large classes of estimators and serve as benchmarks for the assessment of estimation algorithms like the performance measures themselves these bounds are independent of the reference coordinate frame and we discuss their use as system design criteria

optimal designs for bivariate logistic regression summary in drug testing experiments the primary responses of interest are efficacy and toxicity these can be modeled as a bivariate quantal response using the gumbel model for bivariate logistic regression d optimal and q optimal experimental designs are developed for this model the q optimal design minimizes the average asymptotic prediction variance of p l 0 d the probability of efficacy without toxicity at dose d over a desired range of doses the robustness of these designs to parameter misspecification is discussed in addition d efficiencies of q optimal designs and q efficiencies of doptimal designs are presented an extension of the general equivalence theorem to the multivariate case is applied to these designs

bayesian design of experiments using approximate coordinate exchange abstractthe construction of decision theoretical bayesian designs for realistically complex nonlinear models is computationally challenging as it requires the optimization of analytically intractable expected utility functions over high dimensional design spaces we provide the most general solution to date for this problem through a novel approximate coordinate exchange algorithm this methodology uses a gaussian process emulator to approximate the expected utility as a function of a single design coordinate in a series of conditional optimization steps it has flexibility to address problems for any choice of utility function and for a wide range of statistical models with different numbers of variables numbers of runs and randomization restrictions in contrast to existing approaches to bayesian design the method can find multi variable designs in large numbers of runs without resorting to asymptotic approximations to the posterior distribution or expected utility the methodology is demonstrated on

an optimal design problem in rhythmometry a trigonometric regression model is assumed for a problem involving circadian rhythm exhibited by peak expiratory flow experimental designs are sought with a view to estimating a particular nonlinear function of the parameters both optimal and nonoptimal but more practicable designs are derived and their relative efficiencies are established

fast computation of designs robust to parameter uncertainty for nonlinear settings experimental design in nonlinear settings is complicated by the fact that the efficiency of a design depends on the unknown parameter values thus good designs need to be efficient over a range of likely parameter values bayesian design criteria provide a natural framework for achieving such robustness by averaging local design criteria over a prior distribution on the parameters a major drawback to the use of such criteria is the heavy computational burden that they impose we present a clever quadrature scheme that greatly improves the feasibility of using bayesian design criteria we illustrate the method on some designed experiments

curvature adjusted optimal design of sampling times for the inference of pharmacokinetic compartment models in pharmacokinetics compartment models are often used to describe the time course of blood concentration after the administration of a drug in this article we propose an optimal design criterion for precise estimation of parameters included in the compartment model and illustrate the non sequential design of sampling times of blood drug concentration data in individual pharmacokinetics the proposed optimal design criterion minimizes the determinant of the mean squared error matrix of the parameter estimator that is quadratically approximated by the curvature array therefore the proposed criterion considers the intrinsic and parameter effects nonlinearity underlying the compartment model and so is applicable in a pharmacokinetic experiment where the sample size of the blood drug concentration data is quite small

endomorphisms of the cohomology of complex grassmannians for any complex grassmann manifold c we classify all endomorphisms of the rational cohomology ring of c which are nonzero on dimension 2 some applications of this result are given

bayesian optimal design for ordinary differential equation models with application in biological science bayesian optimal design is considered for experiments where the response distribution depends on the solution to a system of nonlinear ordinary differential equations the motivation is an experime

simple and robust experiment design for system identification using fractional models this paper tackles the problems of simple and robust experiment design for system identification using elementary fractional models it is based on a frequency domain approach and allows to determine the best sine input signal maximizing d optimality criterion of the parameters inverse covariance matrix in different contexts first a single parameter any of the parameters of the elementary fractional model is assumed to be unknown next any combination of two and then three parameters are supposed to be unknown finally the problem of robust experiment design is treated when a bounded interval of the estimated parameters is known in the same contexts

robust optimum designs for transformation of the responses in a multivariate chemical kinetic model transformation of both sides of a nonlinear model is often necessary to obtain tractable error distributions but can have a dramatic effect on optimum designs for the parameters of the model this article develops methods for the design of experiments when the value of the power transformation parameter is not precisely known optimum designs for particular transformations together with the efficiency of these designs as the transformation varies are studied for a multivariate model it is shown that by a suitable choice of prior bayesian designs can be found that are robust to the correct transformation a typical equispaced design is evaluated and the properties of designs are studied under various departures from the assumptions

a geometric characterization of optimal designs for regression models with correlated observations we consider the problem of optimal design of experiments for random effects models especially population models where a small number of correlated observations can be taken on each individual whereas the observations corresponding to different individuals are assumed to be uncorrelated we focus on c optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of elfving from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients the theory is demonstrated by finding optimal designs for a linear model with correlated observations and a non linear random effects population model which is commonly used in pharmaco kinetics

optimum experimental designs for multinomial logistic models multinomial responses frequently occur in dose level experiments for example in a study of the influence of gamma radiation on the emergence of house flies musca domestica l 1758 three disjoint outcomes occurred death before the pupae opened death during emergence and life after emergence although the flies are easy to breed this sort of bioassay is in general very expensive since it requires the use of a gamma radiation source experiments therefore need to be designed to involve the minimum number of different doses here the theory of optimum experimental design is applied to provide efficient experiments to estimate the parameters of those multinomial logistic models that are a special case of the multivariate logistic models of glonek and mccullagh 1995 journal of the royal statistical society series b 57 533 546 the purpose is to reduce the overall experimental cost the general equivalence theorem fedorov 1972 theory of optimal experiments is adapted to this class of models providing an effective method of generating and checking the optimality of designs one example on flies demonstrates the method which can be easily implemented

self maps of flag manifolds rationally a map between flag manifolds is seen to be determined up to homotopy by the homomorphism it induces on cohomology two algebraic results for cohomology endomorphisms then serve a to determine those flag manifolds which have nontrivial self maps that factor through a complex projective space and b for a special class of flag manifolds to classify the self maps of their rationalizations up to homotopy 0 introduction the homotopy classes of self maps of a complex projective space are classified by an integral degree and we show that for rational homotopy there is an analogous result for a much larger class of homogeneous spaces these spaces are quotients of a unitary group by a closed connected subgroup of maximal rank and just as for projective spaces the degree of a self map is determined by its effect on a two dimensional cohomology class the first step is to observe that homotopy classes of endomorphisms of the minimal model for the rational homotopy type of such a space correspond bijectively with endomorphisms of the rational cohomology algebra see theorem 1 1 the second step is to classify the cohomology endomorphisms such a classification was given in gh1 for the grassmann manifold of complex p planes in complex n space for n greater than 2p2 and it follows that the rational homotopy classes of self maps are indeed classified by a rational degree here in theorems 1 3 and 1 4 we give the analogous result for the flag manifolds m 1 p q u 1 p q u 1 x u p x u q the cohomology endomorphisms fall into a finite number often one of families each of which is classified by a rational degree some of these families are represented by the endomorphisms induced by conjugation in u 1 p q by elements of the normalizer of u 1 x u p x u q but if 1 p and q are distinct we get only the identity in this way the rest are represented by idempotents which we call projective endomorphisms because they factor through the cohomology of complex projective space such projective endomorphisms for any flag manifold are in turn classified in theorem 1 3 by factorizations over z of the polynomial

fibrations of classifying spaces we investigate fibrations of the form z y x where two of the three spaces are classifying spaces of compact connected lie groups we obtain certain finiteness conditions on the third space which make it also a classifying space our results allow to express some of the basic notions in group theory in terms of homotopy theory i e in terms of classifying spaces as an application we prove that every retract of the classifying space of a compact connected lie group is again a classifying space

parameter neutral optimum design for non linear models some bayesian approaches to d optimum design of experiments are considered from the viewpoint of invariance under reparameterization of the underlying statistical model an invariant criterion is proposed which does not require the detailed specification of a prior and which is shown to be equivalent to g optimality under a jeffreys prior the methods are applied and discussed in the contexts of exponential decay and quantal response models

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

hermitian weighted composition operators on h 2 weighted composition operators have been related to products of composition operators and their adjoints and to isometries of hardy spaces in this paper we identify the hermitian weighted composition operators on h 2 and compute their spectral measures some relevant semigroups are studied the resulting ideas can be used to find the polar decomposition the absolute value and the aluthge transform of some composition operators on h 2

maps between p completed classifying spaces ii we investigate maps between p completed classifying spaces of compact connected lie groups let g and g be two connected compact lie groups for a space x let xp be a p completion of x if p does not divide the order of the weyl group of g we give descriptions of the set of homotopy classes bg p bg p in terms of k theory and in terms of admissible maps of adams and mahmud

experimental design for estimating integrals by numerical quadrature with applications to pharmacokinetic studies many experimental situations including bioavailability studies require the estimation of integrals by numerical quadrature applied to dependent variable observations with measurement error a strategy is described for selecting values for the independent variable e g time the strategy minimizes the expectation of the square of the difference between the exact integral and the quadrature approximation this approach was applied to simulated pharmacokinetic problems including the estimation of bioavailability results indicate that the procedure is potentially useful in reducing the variance of resulting estimates and that it appears to be robust with respect to prior assumptions about model parameter values

locally d optimal designs based on a class of composed models resulted from blending emax and one compartment models a class of nonlinear models combining a pharmacokinetic compartmental model and a pharmacodynamic emax model is introduced the locally d optimal ld design for a 4 parameter composed model is found to be a saturated 4 point uniform ld design with the two boundary points of the design space in the ld design support for a 5 parameter composed model a su cient condition for the ld design to require the minimum number of sampling time points is derived robust ld designs are also investigated for both models it is found that an ld design with k parameters is equivalent to an ld design with k i1 parameters if the linear parameter in the two composed models is a nuisance parameter assorted examples of ld designs are presented research supported by national science foundation nsf grants dms 0103727 dms 0603761 and national institutes of health nih grant p50 at00155 jointly supported by national center for complementary and alternative medicine the o ce of dietary supplements the o ce of research on women s health and national institute of general medicine any opinions flndings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily re ect the views of the nsf and the nih

on operators with similar positive parts abstract let s u s s and t u t t be the polar decompositions of s and t in l h respectively we say that s and t have similar positive parts if s and t are similar in this paper we investigate properties of operators that are preserved under this similarity condition we describe the form for the polar decomposition of an operator when the operator and its adjoint have similar positive parts for this case we also study the existence of invariant subspaces under this assumption

some results on d optimal designs for nonlinear models with applications sufficient conditions are established for the locally d optimal design for a nonlinear model to have a minimal number of support points the conditions are applied to obtain locally d optimal designs for a one compartment pharmacokinetic model and a poisson regression model

rational equivalences between classifying spaces the paper contains a homotopy classification of rational equivalences between classifying spaces of compact connected lie groups with an application to genus sets of such spaces

optimal design for linear interpolation of curves non parametric procedures are often used for the analysis of pharmacokinetic trials fewer design procedures are available for non parametric estimation than for parametric estimation linear interpolation is widely used for curve estimation in pharmacokinetic trials where often only sparse sampling is feasible current design procedures for smoothing or local fit are not suitable as they are based on asymptotic properties and the bias of the estimate is ignored this paper proposes optimal designs that minimize the mean squared error of linear interpolation optimal designs for three situations are considered the first situation is single curve estimation based on an ordinary non linear model the second is estimating several curves in a non linear mixed model setting using an average mean squared error as the design criterion the third situation is destructive sampling where estimating the average curve is the main purpose in the first situation the design results in the best linear interpolation when the variance is constant for the destructive sampling design an algorithm based on approximations is proposed this algorithm can be programmed in a common statistical package numerical examples are used to illustrate the design procedure

homotopy classification of self maps of bg via g actions to complete the classification of self maps of bg theorem 4 2 it remains to prove theorem 4 1 i e to show that the higher limits of certain functors n 1h 2 f p g z p wall vanish and that h0 mp g f i has the expected value in this section these limits h mp g hi will be calculated in the easy cases when g is a simply connected classical group i e one of the groups su n sp n or spin n or when i 2 the computation of h mp g hi for the simply connected exceptional lie groups will be put off until section 6 we begin with some general properties of higher limits let 6 be any small discrete category it is convenient when dealing with higher limits of functors on to let 4 moldenote the category of contravariant functors from v to ad then for any f e w muok we can identify

homotopy fixed point methods for lie groups and finite loop spaces a loop space x is by definition a triple x bx e in which x is a space bx is a connected pointed space and e x qbx is a homotopy equivalence from x to the space qbx of based loops in bx we will say that a loop space x is finite if the integral homology h x z is finitely generated as a graded abelian group i e if x appears at least homologically to be a finite complex in this paper we prove the following theorem

on adic genus and lambda rings sufficient conditions on a space are given which guarantee that the k theory ring is an invariant of the adic genus an immediate consequence of this result about adic genus is that for any positive integer n the power series ring z x 1 x n admits uncountably many pairwise non isomorphic a ring structures

optimum experimental designs for properties of a compartmental model summary three properties of interest in bioavailability studies using compartmental models are the area under the concentration curve the maximum concentration and the time to maximum concentration methods are described for finding designs that minimize the variance of the estimates of these quantities in such a model these methods use prior information both prior estimates and prior distributions are used the designs for an open one compartment model are compared with the corresponding d optimum design for all parameters and also with designs that minimize the sum of the scaled variances of the individual properties

optimal design in random effects regression models summary an approach is proposed to optimal design of experiments for estimating randomeffects regression models the population designs are defined by the number of subjects and the individual designs to be performed cost functions associated with individual designs are incorporated for a given maximal cost an algorithm is proposed for finding the statistical population design that maximises the determinant of the fisher information matrix of the population parameters the fisher information matrix is formulated for linear models and normal distributions the approach is applied to the design of an optimal experiment in toxicokinetics using a first order linearisation of the model several cost functions and designs of various orders are studied an example illustrates the optimal population designs and the increased efficiency of some optimal designs over more standard designs

the classification of 2 compact groups we prove that any connected 2 compact group is classified by its 2 adic root datum and in particular the exotic 2 compact group di 4 constructed by dwyer wilkerson is the only simple 2 compact group not arising as the 2 completion of a compact connected lie group combined with our earlier work with moller and viruel for p odd this establishes the full classification of p compact groups stating that up to isomorphism there is a one to one correspondence between connected p compact groups and root data over the p adic integers as a consequence we prove the maximal torus conjecture giving a one to one correspondence between compact lie groups and finite loop spaces admitting a maximal torus our proof is a general induction on the dimension of the group which works for all primes it refines the andersen grodal moller viruel methods by incorporating the theory of root data over the p adic integers as developed by dwyer wilkerson and the authors furthermore we devise a different way of dealing with the rigidification problem by utilizing obstruction groups calculated by jackowski mcclure oliver in the early 1990s less

the equivalence of constrained and weighted designs in multiple objective design problems abstract several competing objectives may be relevant in the design of an experiment the competing objectives may not be easy to characterize in a single optimality criterion one approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria an alternative approach has been to optimize one criterion subject to constraints on the other criteria an equivalence theorem is presented for the bayesian constrained design problem equivalence theorems are essential in verifying optimality of proposed designs especially when as in most nonlinear design problems numerical optimization is required this theorem is used to show that the results of cook and wong on the equivalence of the weighted and constrained problems apply much more generally the results are applied to bayesian nonlinear design problems with several objectives

a geometric approach to optimal design for one parameter non linear models summary a geometric framework for constructing optimal bayesian designs and maximin designs for non linear models with a single unknown parameter and a prior distribution on that parameter which is restricted in that it comprises exactly two points of support is presented the approach is illustrated by means of selected examples involving logistic regression and the simple exponential model and its applicability to the construction of optimal designs for models with uncontrolled variation and to model robust designs is also demonstrated in addition the method is shown to provide some valuable insights into the general properties of optimal bayesian designs for non linear models

the mean transform of bounded linear operators abstract in this paper we introduce the mean transform of bounded linear operators acting on a complex hilbert space and then explore how the mean transform of weighted shifts behaves in comparison with the aluthge transform

connected finite loop spaces with maximal tori finite loop spaces are a generalization of compact lie groups however they do not enjoy all of the nice properties of compact lie groups for example having a maximal torus is a quite distinguished property actually an old conjecture due to wilkerson says said that every connected finite loop space with a maximal torus is equivalent to a compact connected lie group we give some more evidence for this conjecture by showing that the associated action of the weyl group on the maximal torus always represents the weyl group as a crystallographic group we also develop the notion of normalizers of maximal tori for connected finite loop space and prove for a large class of connected finite loop spaces that a connected finite loop space with maximal tori is equivalent to a compact connected lie group if it has the right normalizer of the maximal torus actually in the cases under consideration the information about the weyl group is sufficient to give the answer all this is done by first studying the analoguous local problems