from jack polynomials to minimal model spectra in this note a deep connection between free field realizations of conformal field theories and symmetric polynomials is presented we give a brief introduction into the necessary prerequisites of both free field realizations and symmetric polynomials in particular jack symmetric polynomials then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods while these results on the representation theory of the minimal models are all known this note exploits the full power of jack polynomials to present significant simplifications of the original proofs in the literature

modularity of relatively rational vertex algebras and fusion rules of principal affine w algebras we study modularity of the characters of a vertex super algebra equipped with a family of conformal structures along the way we introduce the notions of rationality and cofiniteness relative to such a family we apply the results to determine modular transformations of trace functions on admissible modules over affine kac moody algebras and via brst reduction trace functions on minimal series representations of principal affine w algebras

the kazhdan lusztig conjecture for w algebras the main result in this paper is the character formula for arbitrary irreducible highest weight modules of w algebras the key ingredient is the functor provided by quantum hamiltonian reduction which constructs the w algebras from affine kac moody km algebras and in a similar fashion w modules from km modules assuming certain properties of this functor the w characters are subsequently derived from the kazhdan lusztig conjecture for km algebras the result can be formulated in terms of a double coset of the weyl group of the km algebra the hasse diagrams give the embedding diagrams of the verma modules and the kazhdan lusztig polynomials give the multiplicities in the characters

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

quantum reduction for affine superalgebras we extend the homological method of quantization of generalized drinfeld sokolov reductions to affine superalgebras this leads in particular to a unified representation theory of superconformal algebras

the existence of a global fundamental solution for homogeneous hormander operators via a global lifting method we prove the existence of a global fundamental solution x y with pole x for any hormander operator l i 1mxi2 on rn which is homogeneous of degree 2 here homogeneity is meant with respect to a family of non isotropic diagonal maps of the form x 1x1 nxn with 1 1 n due to a global lifting method for homogeneous operators proved by folland comm partial differential equations 2 1977 161 207 there exists a carnot group g and a polynomial surjective map g rn such that l is related to a sub laplacian lg on g we show that it is always possible to perform a global change of variable on g such that the lifting map becomes the projection of g rn rp onto rn we prove that an integration argument over the non compact fibers of provides a fundamental solution for l indeed if g x x y y x y rn x y rp is the fundamental solution of lg we show that g x 0 y y is always integrable with respect to y rp and its y integral is a fundamental solution for l

kac wakimoto characters and universal mock theta functions in recent work bringmann and ono answer a question of kac and show that character formulas for sl r 1 1 modules due to kac and wakimoto are holomorphic parts of nonholomorphic modular functions here we confirm a speculation of ono that these characters are up to a simple q series the universal mock theta functions g 2 w q and g 3 w q of gordon and mcintosh using recent work of bringmann ono kang zwegers and gordon mcintosh we show that g 2 w q and g 3 w q are up to classical theta functions and products the characters of kac and wakimoto as a consequence we include a dictionary that gives a character formula for every classical mock theta function of ramanujan as well as subsequent natural generalizations

design and analysis of group sequential tests based on the type i error spending rate function summary lan demets 1983 devised a method of constructing discrete group sequential boundaries by using the type i error spending rate function it is extended so as to generate asymmetric as well as symmetric two sided boundaries for clinical trials the design aspect of this procedure is explored in terms of the maximum sample size needed to achieve a desired level of power and the expected stopping times under both null and alternative hypotheses finally these properties are employed in search of appropriate type i error spending rate functions for differing situations

quantized enveloping algebras for borcherds superalgebras the rst author was supported in part by nsf grant dms 9300523 the second author was supported in part by the nondirected research fund korea research foundation 1996 the third author was supported in part by a faculty research grant from st lawrence university

sequential testing in clinical trials with repeated measurements summary in some clinical trials the number of subjects on each treatment is fixed and at successive equally spaced time points each subject yields a measurement which reflects response to treatment a model is presented for such repeated measurements data a sequential procedure is proposed for testing the null hypothesis of no difference in mean effect between two treatments in this procedure the maximum number of measurements per subject and absorption probabilities for intermediate significance tests are preassigned to yield a required approximate overall significance level the jth significance test follows the jth measurement on each subject and the test depends on values of model parameters estimated from all the data available so far simulations show that the proposed procedure yields overall significance levels close to 5 when required for data from a variety of

closed testing procedures for group sequential clinical trials with multiple endpoints summary a simple approach is given for conducting closed testing in clinical trials with multiple end points in which group sequential monitoring is planned the approach allows a flexible stopping time the earliest and latest stopping times are described the paradigm is applicable both to clinical trials with multiple endpoints and to the one sided multiple comparison problem of several treatments versus a control the approach leads to enhancements of previous methods and suggestions for new methods an example of a respiratory disease trial with four endpoints is given

on the superconformal index of argyres douglas theories we conjecture a closed form expression for the schur limit of the superconformal index of two infinite series of argyres douglas ad superconformal field theories scfts the a1 a2n 3 and the a1 d2n theories while these scfts can be realized at special points on the coulomb branch of certain n 2 gauge theories their superconformal r symmetries are emergent and hence their indices cannot be evaluated by localization instead we construct the a1 a2n 3 and a1 d2n indices by using a relation to two dimensional qdeformed yang mills theory and data from the class s construction our results generalize the indices derived from the torus partition functions of the two dimensional chiral algebras associated with the a1 a3 and a1 d4 scfts as checks of our conjectures we study the consistency of our results with an s duality recently discussed by us in collaboration with giacomelli and papageorgakis we reproduce known higgs branch relations we check consistency with a series of renormalization group flows and we verify that the small s 1 limits of our indices reproduce expected cardy like behavior we will discuss the s 1

detecting treatment differences in group sequential longitudinal studies with covariate adjustment in longitudinal studies comparing two treatments over a series of common follow up measurements there may be interest in determining if there is a treatment difference at any follow up period when there may be a non monotone treatment effect over time to evaluate this question jeffries and geller 2015 examined a number of clinical trial designs that allowed adaptive choice of the follow up time exhibiting the greatest evidence of treatment difference in a group sequential testing setting with gaussian data the methods are applicable when a few measurements were taken at prespecified follow up periods here we test the intersection null hypothesis of no difference at any follow up time versus the alternative that there is a difference for at least one follow up time results of jeffries and geller 2015 are extended by considering a broader range of modeled data and the inclusion of covariates using generalized estimating equations testing procedures are developed to determine a set of follow up times that exhibit a treatment difference that accounts for multiplicity in follow up times and interim analyses

affine lie algebras and product sum identities abstract we provide four different decompositions for a special infinite product first studied by i schur our product sum decomposition gives two more correspondences between subsets of partitions with parts congruent to 1 or 5 modulo 6 our method uses a special vertex representation of affine lie algebra c 3 1 and admissible representations of a 1 1 as well as quintuple product identity

graded lie superalgebras and the superdimension formula in this paper we investigate the structure of graded lie superalgebras l a a l a where is a countable abelian semigroup and a is a countable abelian group with a coloring map satisfying a certain finiteness condition given a denominator identity for the graded lie superalgebra l we derive a superdimension formula for the homogeneous subspaces l a a a which enables us to study the structure of graded lie superalgebras in a unified way we discuss the applications of our superdimension formula to free lie superalgebras generalized kac moody superalgebras and monstrous lie superalgebras in particular the product identities for normalized formal power series are interpreted as the denominator identities for free lie superalgebras we also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of monstrous lie superalgebras

quantum reduction and representation theory of superconformal algebras abstract we study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction as an application we obtain in a unified way free field realizations and determinant formulas for all superconformal algebras

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

a general statistical principle for changing a design any time during the course of a trial a general method is presented that allows the researcher to change statistical design elements such as the residual sample size during the course of an experiment to include an interim analysis for early stopping when no formal rule for early stopping was foreseen to increase or reduce the number of planned interim analyses to change time points and the type i error spending function for the further design of interim analyses or to change the test statistic the outcome measure etc at the time of a pre planned interim analysis for early stopping or at any time of an interim look without spending part of the type i error level the method offers the option to completely redesign the remaining part of the trial without affecting the type i error level the method is described in the usual brownian motion model and extended to the general context of statistical decision functions it is based on the conditional rejection probability of a decision variable

discrete sequential boundaries for clinical trials summary pocock 1977 o brien fleming 1979 and slud wei 1982 have proposed different methods to construct discrete sequential boundaries for clinical trials these methods require that the total number of decision times be specified in advance in the present paper we propose a more flexible way to construct discrete sequential boundaries the method is based on the choice of a function a t which characterizes the rate at which the error level ac is spent the boundary at a decision time is determined by a t and by past and current decision times but does not depend on the future decision times or the total number of decision times

evaluation of experiments with adaptive interim analyses summary a general method for statistical testing in experiments with an adaptive interim analysis is proposed the method is based on the observed error probabilities from the disjoint subsamples before and after the interim analysis formally an intersection of individual null hypotheses is tested by combining the twop values into a global test statistic stopping rules for fisher s product criterion in terms of critical limits for the p value in the first subsample are introduced including early stopping in the case of missing effects the control of qualitative treatment stage interactions is considered a generalization to three stages is outlined the loss of power when using the product criterion instead of the optimal classical test on the whole sample is calculated for the test of the mean of a normal distribution depending on increasing proportions of the first subsample in relation to the total sample size an upper bound on the loss of power due to early stopping is derived a general example is presented and rules for assessing the sample size in the second stage of the trial are given the problems of interpretation and precautions to be taken for applications are discussed finally the sources of bias for estimation in such designs are described

hierarchical testing of multiple endpoints in group sequential trials we consider the situation of testing hierarchically a key secondary endpoint in a group sequential clinical trial that is mainly driven by a primary endpoint by mainly driven we mean that the interim analyses are planned at points in time where a certain number of patients or events have accrued on the primary endpoint and the trial will run either until statistical significance of the primary endpoint is achieved at one of the interim analyses or to the final analysis we consider both the situation where the trial is stopped as soon as the primary endpoint is significant as well as the situation where it is continued after primary endpoint significance to further investigate the secondary endpoint in addition we investigate how to achieve strong control of the familywise error rate fwer at a pre specified significance level alpha for both the primary and the secondary hypotheses we systematically explore various multiplicity adjustment methods starting point is a naive strategy of testing the secondary endpoint at level alpha whenever the primary endpoint is significant hung et al j biopharm stat 2007 17 1201 1210 have already shown that this naive strategy does not maintain the fwer at level alpha we derive a sharp upper bound for the rejection probability of the secondary endpoint in the naive strategy this suggests a number of multiple test strategies and also provides a benchmark for deciding whether a method is conservative or might be improved while maintaining the fwer at alpha we use a numerical example based on a real case study to illustrate the results of different hierarchical test strategies

on infinite root systems we define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a kac moody lie algebra we study these objects from a purely formal and geometrical point of view as well as in relation to their associated lie algebras this leads to a coherent theory of root systems bases subroot systems lie algebras defined by root data and subalgebras

fock representations and brst cohomology insl 2 current algebra we investigate the structure of the fock modules overa 1 1 introduced by wakimoto we show that irreducible highest weight modules arise as degree zero cohomology groups in a brst like complex of fock modules chiral primary fields are constructed as brst invariant operators acting on fock modules as a result we obtain a free field representation of correlation functions of thesu 2 wzw model on the plane and on the torus we also consider representations of fractional level arising in polyakov s 2d quantum gravity finally we give a geometrical borel weil like interpretation of the wakimoto construction

level rank duality in nonunitary coset theories it is shown that the conformal field theories su n k su n 1 su n k 1 and su m l su m 1 su m 1 1 are equivalent where k n n m l m m n for a pair m n of relatively prime positive integers

sequential comparison of changes with repeated measurements data abstract there are many clinical trials in which the patients enter sequentially and a response variable is measured repeatedly over time for each patient a group sequential procedure is proposed for comparing the rates of change between two treatment groups some existing procedures for testing the equality of means between two treatment groups with repeated measurements data such as those proposed by armitage stratton and worthington and by geary can be interpreted as special cases of the proposed procedure under a linear mixed effects model the asymptotic joint distribution of the sequentially computed statistics is derived construction of the group sequential boundaries is based on this distribution theory the extension to nonnormal distributions of errors is also discussed the proposed procedure allows for using either the sequential method of slud and wei or that of lan and demets this procedure is then applied to repeated bone density measurements on a sample of 74 middle aged women and

representation formulae of solutions of second order elliptic inequalities abstract we prove various representation formulae of solutions of second order differential inequalities on r n as an outcome we obtain positivity results of the solutions of l u 0 where l is a general second order elliptic operator in divergence form results of this type are very useful for studying related liouville theorems of second order semilinear inequalities

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

fundamental solution of a higher step grushin type operator abstract we examine a class of grushin type operators p k where k n 0 defined in 1 1 the operators p k are non elliptic and degenerate on a sub manifold of r n l geometrically they arise via a submersion from a sub laplace operator on a nilpotent lie group of step k 1 we explain the geometric framework and prove some analytic properties such as essential self adjointness the main purpose of the paper is to give an explicit expression of the fundamental solution of p k our methods rely on an appropriate change of coordinates and involve the theory of bessel and modified bessel functions together with weber s second exponential integral

vertex operators of admissible modules ofuq c 1 n using our recent bosonic realization ofuq sp2n we construct explicitly the vertex operators for the level 1 2 modules ofuq sp2n using bosonic fields our method contains a detailed analysis of all theq intertwining relations

longitudinal clinical trials with adaptive choice of follow up time in longitudinal studies comparing two treatments with a maximum follow up time there may be interest in examining treatment effects for intermediate follow up times one motivation may be to identify the time period with greatest treatment difference when there is a non monotone treatment effect over time another motivation may be to make the trial more efficient in terms of time to reach a decision on whether a new treatment is efficacious or not here we test the composite null hypothesis of no difference at any follow up time versus the alternative that there is a difference at at least one follow up time the methods are applicable when a few measurements are taken over time such as in early longitudinal trials or in ancillary studies suppose the test statistic z t k will be used to test the hypothesis of no treatment effect at a fixed follow up time t k in this context a common approach is to perform a pilot study on n1 subjects and evaluate the treatment effect at the fixed time points t1 t k and choose t as the value of t k for which z t k is maximized having chosen t a second trial can be designed in a setting with group sequential testing we consider several adaptive alternatives to this approach that treat the pilot and second trial as a seamless combined entity and evaluate type i error and power characteristics the adaptive designs we consider typically have improved power over the common separate trial approach

sequential monitoring for comparison of changes in a response variable in clinical studies the spending function approach proposed by lan and demets 1983 biometrika 70 659 663 for sequential monitoring of clinical trials is applied to situations where comparison of changes in a continuous response variable between two groups is the primary concern death loss to follow up and missed visits could cause follow up measurements to be right censored or missing for some participants furthermore the probability of being censored may be dependent on the parameter value of the response variable informative censoring we propose to compare treatment effects by comparing areas under the expected response change curves between the two groups when the response curves are linear as a function of time in both groups this comparison is equivalent to comparing the rates of change in the response variable covariances of the sequential test statistics are derived conditions for having independent increments are presented for studies designed to evaluate long term treatment effects spending functions obtained by shifting the usual spending functions kim and demets 1987 biometrika 74 149 154 to the right and then rescaling to the remaining interval are also proposed such a shifted spending function is applied to the monitoring plan for the lung health study anthonisen 1989 american review of respiratory diseases 140 871 872

characters and fusion rules for w algebras via quantized drinfeld sokolov reduction using the cohomological approach tow algebras we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized drinfeld sokolov reduction

punctures and p spin curves from matrix models this article investigates the intersection numbers of the moduli space of p spin curves with the help of matrix models the explicit integral representations that are derived for the generating functions of these intersection numbers exhibit p stokes domains labelled by a spin component l taking values l 1 0 1 2 p 2 earlier studies concerned integer values of p but the present formalism allows one to extend our study to half integer or negative values of p which turn out to describe new types of punctures or marked points on the riemann surface they fall into two classes ramond l 1 absent for positive integer p and neveu schwarz l ne 1 the intersection numbers of both types are computed from the integral representation of the n point correlation functions in a large n scaling limit we also consider a supersymmetric extension of the random matrix formalism to show that it leads naturally to an additional logarithmic potential open boundaries on the surface or admixtures of r and ns punctures may be handled by this extension

ramanujan s lost notebook and the virasoro algebra by using the theory of vertex operator algebras we gave a new proof of the famous ramanujan s modulus 5 modular equation from his lost notebook p 139 in r furthermore we obtained an infinite list of q identities for all odd moduli thus we generalized the result of ramanujan

relaxed highest weight modules i rank 1 cases relaxed highest weight modules play a central role in the study of many important vertex operator super algebras and their associated logarithmic conformal field theories including the admissible level affine models indeed their structure and their super characters together form the crucial input data for the standard module formalism that describes the modular transformations and grothendieck fusion rules of such theories in this article character formulae are proved for relaxed highest weight modules over the simple admissible level affine vertex operator superalgebras associated to mathfrak s mathfrak l 2 and mathfrak osp 1 vert 2 moreover the structures of these modules are specified completely this proves several conjectural statements in the literature for mathfrak s mathfrak l 2 at arbitrary admissible levels and for mathfrak osp 1 vert 2 at level frac 5 4 for other admissible levels the mathfrak osp 1 vert 2 results are believed to be new

n point and higher genus osp 1 2 fusion we study affine osp 1 2 fusion the fusion in osp 1 2 conformal field theory for example higher point and higher genus fusion is discussed the fusion multiplicities are characterized as discretized volumes of certain convex polytopes and are written explicitly as multiple sums measuring those volumes we extend recent methods developed to treat affine su 2 fusion they are based on the concept of generalized berenstein zelevinsky triangles and virtual couplings higher point tensor products of finite dimensional irreducible osp 1 2 representations are also considered the associated multiplicities are computed and written as multiple sums

conformal field theories via hamiltonian reduction constraining thesl 3 wzw model we construct a reduced theory which is invariant with respect to the new chiral algebraw 3 2 this symmetry is generated by the stress energy tensor two bosonic currents with spins 3 2 and theu 1 current we conjecture a kac formula that describes the highly reducible representation for this algebra we also discuss the quantum hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras

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

two sample repeated significance tests based on the modified wilcoxon statistic abstract the asymptotic distribution theory of sequentially computed modified wilcoxon scores is developed for two sample survival data with random staggered entry and random loss to follow up the asymptotic covariance indicates generally dependent modified wilcoxon increments contradicting the authors reading of jones and whitehead 1979 a repeated significance testing procedure is presented for testing the equality of two survival distributions based on the asymptotic theory the early stopping properties of this procedure are illustrated by a prostate cancer example

bosonic realizations ofuq c 1 n abstract we construct explicitly the quantum symplectic affine algebrauq sp 2n using bosonic fields the fock space decomposes into irreducible modules of level 1 2 quantizing the feingold frenkel construction forq 1

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

representation formulae and inequalities for solutions of a class of second order partial differential equations let l be a possibly degenerate second order differential operator and let n d 2 q be its fundamental solution at here d is a suitable distance in this paper we study necessary and sufficient conditions for the weak solutions of lu f u 0 on r n to satisfy the representation formula r u r n f u we prove that r holds provided f is superlinear without any assumption on the behavior of u at infinity on the other hand if u satisfies the condition lim inf f r d 2r u d 0 then r holds with no growth assumptions on f

designed extension of studies based on conditional power we propose a flexible method of extending a study based on conditional power the possibility for extension when the p value at the planned end is small but not statistically significant is built in to the design of the study the significance of the treatment difference at the planned end is used to determine the number of additional observations needed and the critical value necessary for use after accruing those additional observations it may therefore be thought of as a two stage procedure even though the observed treatment difference at stage 1 is used to make decisions the type i error rate is protected

weyl kac type character formula for admissible representations of borcherds kac moody lie superalgebras we prove that borcherds kac moody lie superalgebras yield weyl kac type character formulas for some specific types of admissible representations our results generalize the results of kac and wakimoto proc natl acad sci usa 85 4956 4960 1988 by extending the weyl kac character formula for integrable irreducible highest weight modules over kac moody algebras to the case of admissible irreducible highest weight modules over borcherds kac moody lie superalgebras