modular invariant representations of infinite dimensional lie algebras and superalgebras abstract in this paper we launch a program to describe and classify modular invariant representations of infinite dimensional lie algebras and superalgebras we prove a character formula for a large class of highest weight representations l of a kac moody algebra unk with a symmetrizable cartan matrix generalizing the weyl kac character formula kac v g 1974 funct anal appl 8 68 70 in the case of an affine unk this class includes modular invariant representations of arbitrary rational level m t u where t unk z and u unk n are relatively prime and m g g u g is the dual coxeter number we write the characters of these representations in terms of theta functions and calculate their asymptotics generalizing the results of kac and peterson kac v g peterson d h 1984 adv math 53 125 264 and of kac and wakimoto kac v g wakimoto m 1988 adv math 70 156 234 for the u 1 integrable case we work out in detail the case unk a1 1 in particular classifying all its modular invariant representations furthermore we show that the modular invariant representations of the virasoro algebra vir are precisely the minimal series of belavin et al belavin a a polyakov a m zamolodchikov a b 1984 nucl phys b 241 333 380 using the character formulas of feigin and fuchs feigin b l fuchs d b 1984 lect notes math 1060 230 245 we show that tensoring the basic representation and modular invariant representations of a1 1 produces all modular invariant representations of vir generalizing the results of goddard et al goddard p kent a olive d 1986 commun math phys 103 105 119 and of kac and wakimoto kac v g wakimoto m 1986 lect notes phys 261 345 371 in the unitary case we study the general branching functions as well all these results are generalized to the kac moody superalgebras introduced by kac kac v g 1978 adv math 30 85 136 and to n 1 super virasoro algebras we work out in detail the case of the superalgebra b 0 1 1 showing in particular that restricting to its even part produces again all modular invariant representations of vir these results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations

on osp 2 2 conformal field theories we study the conformal field theories corresponding to current superalgebras osp 2 2 1 k and osp 2 2 2 k we construct the free field realizations screen currents and primary fields of these current superalgebras at general level k all the results for osp 2 2 2 k are new and the results for the primary fields of osp 2 2 1 k also seem to be new our results are expected to be useful in the supersymmetric approach to gaussian disordered systems such as the random bond ising model and the dirac model

character formulas for irreducible modules of the lie superalgebras sl m n kac distinguished between typical and atypical finite dimensional irreducible representations of the lie superalgebras sl m n and provided an explicit character formula appropriate to all the typical representations here the range of validity of some character formulas for atypical representations that have been proposed are discussed several of them are of the kac weyl type but then it is proved that all formulas of this type fail to correctly give the character of one particular atypical representation of sl 3 4 having ruled out therefore all such formulas a completely new extension of the kac weyl character formula is proposed the validity of this formula in the case of all covariant tensor irreducible representations is proved and some evidence in support of the conjecture that it covers all irreducible representations of sl m n is presented

optimal item difficulty for the three parameter normal ogive response model in tailored testing it is important to determine the optimal difficulty of the next item to present to the examinee this paper shows that the difference that maximizes information for the three parameter normal ogive response model is approximately 1 7 times the optimal difference b for the three parameter logistic model under the normal model calculation of the optimal difficulty for minimizing the bayes risk is equivalent to maximizing an associated information function

correlation functions in liouville theory we show that after integrating over the zero mode in liouville correlation functions the remaining functional integral resembles a free theory and may be evaluated by formally continuing the central charge we apply this technique to the unitary minimal models coupled to gravity on the sphere computing a number of three point functions after taking into account the normalizations of operators and the functional integral we find exact agreement between the liouville three point functions and the results from matrix models

on the statistical analysis of capture experiments summary a procedure is given for estimating the size of a closed population in the presence of heterogeneous capture probabilities using capture recapture data when it is possible to model the capture probabilities of individuals in the population using covariates the results include the estimation of the parameters associated with the model of the capture probabilities and the use of these estimated capture probabilities to estimate the population size confidence intervals for the population size using both the asymptotic normality of the estimator and a bootstrap procedure for small samples are given

test theory and psychometrika the past twenty five years on the occasion ofpsychometrika s fiftieth anniversary the past twenty five years developments in mental test theory are reviewed with special emphasis on the topics receiving attention in the pages of this journal analogous reviews forpsychometrika s first quarter century were given by gulliksen and guilford in 1961 much of the recent progress in test theory and in other branches of psychometrics as well has been made by treating the problems in this field as being essentially ones of statistical inference it is concluded that a research in test theory is in a healthy state and b psychometrika is an important source of information about that research

logistic regression in capture recapture models the effect of population heterogeneity in capture recapture or dual registration models is discussed an estimator of the unknown population size based on a logistic regression model is introduced the model allows different capture probabilities across individuals and across capture times the probabilities are estimated from the observed data using conditional maximum likelihood the resulting population estimator is shown to be consistent and asymptotically normal a variance estimator under population heterogeneity is derived the finite sample properties of the estimators are studied via simulation an application to finnish occupational disease registration data is presented

a general theory for modeling capture recapture data from a closed population a general theory for estimating the size of a closed population from multiple recapture data is presented this theory is easily extended to open population models for multiple recapture data estimation is based on a log linear model developed for modeling dependent capture recapture data when capture probabilities vary temporally with behavioral response and are heterogeneous among animals models for complete capture history data are developed along with the first log linear models for removal data the maximum likelihood estimator of the model parameters along with the estimated covariance matrix are presented

quantum group uq dl singular vectors in the poincare birkhoff witt basis we give explicit expressions for the singular vectors of uq dl in terms of the poincare birkhoff witt basis we relate these expressions to those in terms of the simple root vectors

open capture recapture models with heterogeneity ii jolly seber model summary estimation of abundance is important in both open and closed population capture recapture analysis but unmodeled heterogeneity of capture probability leads to negative bias in abundance estimates this article defines and develops a suite of open population capture recapture models using finite mixtures to model heterogeneity of capture and survival probabilities model comparisons and parameter estimation use likelihood based methods a real example is analyzed and simulations are used to check the main features of the heterogeneous models especially the quality of estimation of abundance survival recruitment and turnover the two major advances in this article are the provision of realistic abundance estimates that take account of heterogenetiy of capture and an appraisal of the amount of overestimation of survival arising from conditioning on the first capture when heterogeneity of survival is present

adjusting for nonresponse bias using logistic regression summary a method of adjustment for nonresponse is considered no information about the nonrespondents is required but the respondents are assumed to answer all questions of interest one or more call backs are assumed the probabilities of response are represented by a logistic regression model in which variables in the survey are explanatory variables the probabilities are estimated using a modified conditional maximum likelihood approach based on the respondents the estimated probabilities are used in a horvitzthompson type estimator to reduce bias in the estimation of sample moments simulations suggest that the method reduces the mean squared error of estimation when response probabilities depend strongly on variables in the survey

unitary representations of non compact supergroups we give a general theory for the construction of oscillator like unitary irreducible representations uirs of non compact supergroups in a super fock space this construction applies to all non compact supergroupsg whose coset spaceg k with respect to their maximal compact subsupergroupk is hermitean supersymmetric we illustrate our method with the example of su m p n q by giving its oscillator like uirs in a particle state basis as well as supercoherent state basis the same class of uirs can also be realized over the super hilbert spaces of holomorphic functions of az variable labelling the coherent states

and as vertex operator extensions of dual affine algebras we discover a realisation of the affine lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with dual levels and an auxiliary level 1 algebra the duality relation between the levels is we construct the representation of on a sum of tensor products of and modules and decompose it into a direct sum over the spectral flow orbit this decomposition gives rise to character identities which we also derive the extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs conversely we show how the representations are constructed from representations

accommodating unmodeled heterogeneity in double observer distance sampling surveys mark recapture models applied to double observer distance sampling data neglect the information on relative detectability of objects contained in the distribution of observed distances a difference between the observed distribution and that predicted by the mark recapture model is symptomatic of a failure of the assumption of zero correlation between detection probabilities implicit in the mark recapture model we develop a mark recapture based model that uses the observed distribution to relax this assumption to zero correlation at only one distance we demonstrate its usefulness in coping with unmodeled heterogeneity using data from an aerial survey of crabeater seals in the antarctic

the use of auxiliary variables in capture recapture and removal experiments the dependence of animal capture probabilities on auxiliary variables is an important practical problem which has not been considered in the development of estimation procedures for capturerecapture and removal experiments in this paper the linear logistic binary regression model is used to relate the probability of capture to continuous auxiliary variables the auxiliary variables could be environmental quantities such as air or water temperature or characteristics of individual animals such as body length or weight maximum likelihood estimators of the population parameters are considered for a variety of models which all assume a closed population testing between models is also considered the models can also be used when one auxiliary variable is a measure of the effort expended in obtaining the sample

maximum empirical likelihood estimation for abundance in a closed population from capture recapture data summarycapture recapture experiments are widely used to collect data needed for estimating the abundance of a closed population to account for heterogeneity in the capture probabilities huggins 1989 and alho 1990 proposed a semiparametric model in which the capture probabilities are modelled parametrically and the distribution of individual characteristics is left unspecified a conditional likelihood method was then proposed to obtain point estimates and wald type confidence intervals for the abundance empirical studies show that the small sample distribution of the maximum conditional likelihood estimator is strongly skewed to the right which may produce wald type confidence intervals with lower limits that are less than the number of captured individuals or even are negative in this paper we propose a full empirical likelihood approach based on huggins and alho s model we show that the null distribution of the empirical likelihood ratio for the abundance is asymptotically chi squared with one degree of freedom and that the maximum empirical likelihood estimator achieves semiparametric efficiency simulation studies show that the empirical likelihood based method is superior to the conditional likelihood based method its confidence interval has much better coverage and the maximum empirical likelihood estimator has a smaller mean square error we analyse three datasets to illustrate the advantages of our empirical likelihood approach

log linear models for capture recapture summary log linear models are developed for capture recapture experiments and their advantages and disadvantages discussed ways in which they can be extended sometimes with only partial success to open populations subpopulations trap dependence and long chains of recapture periods are presented the use of residual patterns and analysis of subsets of data to identify behavioural patterns and acceptable models is emphasised and illustrated with two examples 1 capture recapture the biological problem with which we are concerned is that of estimating the parameters of a free living population of unknown size from a sequence of s samples in each of which any individual either is or is not observed the models in this paper require that each animal be marked naturally or artificially in such a way that we can observe the numbers of animals which have any particular pattern of being or not being observed through the sequence of samples there are t 2s 1 different observable patterns which we shall refer to as capture histories although there is no requirement for physical capture and denote each pattern by a sequence of l s and 2 s representing respectively seen or not seen in a particular sample a general pattern of capture history will be denoted a with ra the number of individuals observed to have this pattern thus for example with s 3 samples and a 121 r 2 is the number of animals seen in the first and third samples but not in the second the pattern with all 2 s is by definition unobservable in general the population will be subject to demographic change by birth or immigration and by death or emigration the simplest assumption for the response of an animal to the observer one which underlies the classical petersen schnabel or jolly seber estimators is that every animal alive in the population at a sampling time has the same probability of being observed this probability may vary from one sample to another or be constant over all or a number of samples if this assumption fails it may do so for either or both of two reasons the first is that of trap dependence the probability of an individual being observed in a sample depending on its past history of capture the second is that of heterogeneity between individuals who because of differences in their behaviour or in where they live inherently have different probabilities of being observed in any sample the basic idea of heterogeneity refers to properties of individuals which remain constant over time but individual animals may also have time varying behaviour the aspect most commonly considered being temporary departure of an individual from the population various excellent texts describe theoretical and practical aspects of capture recapture notably the comprehensive one by seber 1982 the introduction by begon 1979 the

representation theory of the affine lie superalgebra at fractional level n 2 noncritical strings are closely related to the wess zumino novikov witten model and there is much hope to further probe the former by using the algebraic apparatus provided by the latter an important ingredient is the precise knowledge of the representation theory at fractional level in this paper the embedding diagrams of singular vectors appearing in verma modules for fractional values of the level p and q coprime are derived analytically the nilpotency of the fermionic generators in hslc requires the introduction of a nontrivial generalisation of the mff construction to relate singular vectors among themselves the diagrams reveal a striking similarity with the degenerate representations of the n 2 superconformal algebra

binary programming and test design an algorithmic approach to test design using information functions is presented the approach uses a special branch of linear programming i e binary programming in addition results of some benchmark problems are presented within the same framework it is also possible to formulate the problem of individualized testing

higher level appell functions modular transformations and characters we study modular transformation properties of a class of indefinite theta series involved in characters of infinite dimensional lie superalgebras the level l appell functions open image in new window satisfy open quasiperiodicity relations with additive theta function terms emerging in translating by the period generalizing the well known interpretation of theta functions as sections of line bundles the open image in new window function enters the construction of a section of a rank l 1 bundle open image in new window we evaluate modular transformations of the open image in new window functions and construct the action of an sl 2 subgroup that leaves the section of open image in new window constructed from open image in new window invariant

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

bias reduction for multiple recapture estimators of closed population size a bias correction for multiple recapture sampling is proposed the correction is similar to chapman s 1951 university of california publication in statistics 1 131 160 correction of the lincoln petersen estimator of population size small sample properties of the modified population size estimator are examined using monte carlo simulation substantial reductions in estimator bias and variance are obtained with the modified estimator

some practical aspects of a conditional likelihood approach to capture experiments the use of conditional likelihood methods in the analysis of capture data allows the modeling of capture probabilities in terms of observable characteristics of the captured individuals and the trapping occasions the resulting models may then be used to estimate the size of the population here the use of conditional likelihood procedures to construct models for capture probabilities is discussed and illustrated by an example