the induction coefficient of induced representation theory its algebra and an analytic expression the induction coefficient is defined as the element of a matrix reducing the induced representation into irreducible constituents the authors give the algebra of these coefficients and a closed analytic expression in terms of matrix elements of coset representatives the consequences of this choice are derived and the importance of some of these results for the racah wigner coupling algebra of the symmetric and unitary groups noted

recursive calculation of transformation factors in terms of primitive factors the authors present an algorithm for recursively calculating coupling recoupling induction and similar coefficients in terms of the primitive coefficients the algorithm is shown to be complete for any compact group they use the 6 j symbol as an example for describing their algorithm but the algorithm applies to a wide class of group theoretic transformation factors

algebraic solutions for all dihedral groups the significant problem with using point groups is the dependence upon extensively tabulated results we present simple algebraic expressions for the primitive characters matrix irreps symmetry adapted functions safs and clebsch gordan coefficients for all dihedral groups dn cnv dnd and dnh those results for arbitrary n and for single and double valued representations have been derived in a simple manner without using group tables previously incomplete tabulated results are now redundant in particular the parity dependence of the safs of the improper dihedral groups is shown analytically simple relations are derived between the safs of the proper and improper dihedral groups so that the safs of the latter can be easily obtained from the safs of dn

local empirical likelihood inference for varying coefficient density ratio models based on case control data in this article we develop a varying coefficient density ratio model for case control studies the case and control samples come from two different distributions under the model assumption the ratio of the two densities is related to the linear combination of covariates with varying coefficients through a known function a special case is the exponential tilt model where the log ratio of the two densities is a linear function of covariates we propose a local empirical likelihood el approach to estimate the nonparametric coefficient functions under some regularity assumptions the proposed estimators are shown to be consistent and asymptotically normally distributed the sieve empirical likelihood ratio selr test statistic for detecting whether the varying coefficients are really constant and other related hypotheses is constructed and it follows approximately a chi squared distribution we introduce a modified bootstrap procedure to estimate the null distribution of the selr when sample size is s

tail estimation for window censored processes this article develops methods to estimate the tail and full distribution of the lengths of the 0 intervals in a continuous time stationary ergodic stochastic process that takes the values 0 and 1 in alternating intervals the setting is that each of many such 0 1 processes has been observed during a short time window thus the observed 0 intervals could be noncensored right censored left censored or doubly censored and the lengths of 0 intervals that are ongoing at the beginning of the observation window have a length biased distribution we exhibit parametric conditional maximum likelihood estimators for the full distribution develop maximum likelihood tail estimation methods based on a semiparametric generalized pareto model and propose goodness of fit plots finite sample properties are studied by simulation and asymptotic normality is established for the most important case the methods are applied to estimation of the length of off road glances in the 100 car study a big naturalistic driving

subduction coefficients of brauer algebras and racah coefficients of o n and sp 2 m ii racah coefficients irreducible representations of brauer algebras are discussed in the non standard basis a method for evaluating subduction coefficients sdcs i e the transformation coefficients between standard and non standard bases of brauer algebras is outlined non trivial sdcs of for are derived racah coefficients of o n and sp 2m can be derived from subduction coefficients of brauer algebras by using the schur weyl duality relation between and o n or sp 2m

on the linear equation method for the subduction problem in symmetric groups we focus on the transformation matrices between the standard young yamanouchi basis of an irreducible representation for the symmetric group sn and the split basis adapted to the direct product subgroups we introduce the concept of subduction graph and show that it conveniently describes the combinatorial structure of the equation system arisen from the linear equation method thus we can outline an improved algorithm to solve the subduction problem in symmetric groups by a graph searching process we conclude by observing that the general matrix form for multiplicity separations resulting from orthonormalization can be expressed in terms of sylvester matrices relative to a suitable inner product in the multiplicity space

a note on colored homfly polynomials for hyperbolic knots from wzw models using the correspondence between chern simons theories and wess zumino witten models we present the necessary tools to calculate colored homfly polynomials for hyperbolic knots for two bridge hyperbolic knots we derive the colored homfly invariants in terms of crossing matrices of the underlying wess zumino witten model our analysis extends previous works by incorporating non trivial multiplicities for the primaries appearing in the crossing matrices so as to describe colorings of homfly invariants beyond the totally symmetric or anti symmetric representations of su n the crossing matrices directly relate to 6j symbols of the quantum group mathcal u q su n we present powerful methods to calculate such quantum 6j symbols for general n this allows us to determine previously unknown colored homfly polynomials for two bridge hyperbolic knots we give explicitly the homfly polynomials colored by the representation 2 1 for two bridge hyperbolic knots with up to eight crossings yet the scope of application of our techniques goes beyond knot theory e g our findings can be used to study correlators in wess zumino witten conformal field theories or in the limit to classical groups to determine color factors for yang mills amplitudes

algebraic solutions for point groups the tetrahedral group for the group chain t d2 c2 a factorization lemma is discovered for the irreducible symmetry operators iso s of a semidirect product group of two abelian groups by means of it simple algebraic expressions of the t d2 single valued irreducible matrices iso s and symmetry adapted functions are derived using the double induced technique the algebraic solutions for the double valued representations of group t in the group chain t c2 are also obtained the simplicity and elegance of the algebraic solution of the point group can be compared with that of the analytic solution of the rotation group

point group symmetries and gaussian integration abstract a computationally efficient and exact method of symmetrizing a complete set of angular functions using gaussian integration is presented this technique will be useful in situations such as crystal field type calculations full potential electronic structure calculations and wherever symmetrized functions of a given irreducible representation are needed as an example combinations of spherical harmonics transforming as the completely symmetric irreducible representation are given for all the 32 crystallographic point groups in three dimensions

explicit evaluation of coupling coefficients for the most degenerate representations of so n explicit results for particular coupling coefficients for the most degenerate representations of so n are given the isoscalar factors which allow a recursive calculation of arbitrary coupling coefficients from those of so 3 are derived 6j and 9j symbols for most degenerate representations are also briefly discussed

more on coupling coefficients for the most degenerate representations of so n we present explicit closed form expressions for the general group theoretical factor appearing in the topology of a high temperature expansion of so n symmetric lattice models this object which is closely related to 6j symbols for the most degenerate representation of so n is discussed in detail

inequalities and bounds for kernel length biased density estimation in this note non parametric estimates of the length biased probability density function and related reliability measures are presented non parametric estimates are also presented under random censoring inequalities and bounds for the error of kernel estimators used for the estimation of length biased probability densities are obtained non asymptotic bounds and stochastic convergence results are established inference for length biased energy functions is developed and implemented

racah wigner algebra for q deformed algebras the concepts of vector coupling coefficients and recoupling coefficients are generalized to q deformed algebras their properties under complex conjugation and permutation of irreps are derived relations between the coefficients for example the racah backcoupling rule are proved it is shown that these properties may be used to recursively calculate the coupling and recoupling coefficients for all q algebras the 3 jm and 6 j symbols for su 2 q are used to illustrate the building up method

a class of 6 j symbols for so 2l 1 in terms of rotation matrices for so 3 it is shown that a 6 j symbol for so 2l 1 in which four primitive spinor representations 1 21 2 1 2 appear is directly related to an so 3 rotation matrix possessing a rank of l 1 2 and characterised by the euler angles 0 1 2 pi 0 an interpretation is given in terms of a rotation operator exp 1 2i pi ty acting in the combined spin and quasispin space of an atomic l shell whose states mod t mt are defined in a quasiparticle scheme in which four coupled spinors 1 21 2 1 2 are used

split bases and multiplicity separations in symmetric group transformation coefficients we consider matrices transforming between the standard young yamanouchi basis of the symmetric group and bases adapted to the product subgroups the split basis we derive closed formulae for transformation coefficients for b 3 which includes the first cases when a choice of multiplicity separation is required we discuss considerations which can be applied to obtain a simple form for the multiplicity separation we show that the combinatorial and algebraic structure of the littlewood richardson rule also known as the biedenharn louck pattern calculus does not assist with finding a simple multiplicity separation

6j symbols for symmetric representations of so n as the double series the corrected triple sum expression of alisauskas 1987 j phys a math gen 20 35 for the recoupling racah coefficients 6j symbols of the symmetric most degenerate representations of the orthogonal groups so n previously derived from the fourfold sum expression of alisauskas also related to the result of hormes and junker 1999 j phys a math gen 32 4249 is rearranged into three new different double sum expressions related to the hypergeometric kampe de feriet type series and a new triple sum expression with preferable summation condition the regge type symmetry of special 6j symbols of the orthogonal groups so n in terms of special kampe de feriet f1 41 3 series is revealed the recoupling coefficients for antisymmetric representations of symplectic group sp 2n are derived using their relation with the recoupling coefficients of the formal orthogonal group so 2n

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

statistical models for intercepted data abstract consider a population of replaceable items of different ages and suppose that for each item there is a certain measurable attribute that is a monotone function perhaps random of the item s age to fix ideas we refer to this attribute as deterioration and we assume that as items age their level of deterioration increases we further assume that as the level of deterioration hits a certain threshold value perhaps unknown the item dies and disappears from the population so it is not available for sampling any more since items that deteriorate at a slower rate live longer any sample obtained over a short period of time would overrepresent slowly deteriorating items consequently under such circumstances longitudinal studies are often needed to statistically study deterioration as a function of age this article presents a statistical model that enables us to study the deterioration function from a single cross section snapshot of the population or some random parts of it specifically

the clebsch gordan coefficient for su m n gel fand basis the outer product reduction coefficients orc of the graded state permutation group which differ from the orc of the ordinary permutation group only in signs have been identified with the clebsch gordan coefficients cgc for the special gel fand basis of the graded unitary group su m n the cgc for a general gel fand basis of su m n can be easily obtained from those for the special one tables of the su m n cgc are presented which are valid for arbitrary m and n

parametric estimation for window censored recurrence data many applications in particular the failure repair and replacement of industrial components or physical infrastructure involve recurrent events frequently the available data are window censored only events that occurred during a particular interval are recorded window censoring presents a challenge for recurrence data analysis for statistical inference from window censored recurrence data we derive the likelihood function for a model in which the distributions of inter recurrence intervals in a single path need not be identical and may be associated with covariate information we assume independence among different sample paths we propose a distribution to model the effect of external interventions on recurrence processes this distribution can represent a phenomenon frequently observed in practice that the probability of process regeneration increases with the number of historical interventions for example an item that had a given number of repairs is generally more likely to be replaced i

an information matrix test for logistic regression models based on case control data summary we propose an information matrix based goodness of fit statistic to test the validity of the logistic regression model based on case control data by extending the information matrix test of white 1982 for detecting one sample parametric model misspecification to the semiparametric profile likelihood setting under a two sample semiparamnetric model which is equivalent to the assumed logistic regression model the proposed test statistic requires a high dimensional matrix inversion but is otherwise easily computed and has an asymptotic chi squared distribution this test statistic is an alternative to the kolmogorov smirnov type statistic of qin zhang 1997 and the chi squared type statistic of zhang 1999 and needs neither to employ a bootstrap method to evaluate its critical values nor to group the combined sample data into a finite number of mutually exclusive categories even when the underlying population distribution is continuous we demonstrate that the proposed test statistic and its asymptotic distribution may be obtained by fitting the prospective logistic regression model to case control data we present some results on simulation and on the analysis of three real datasets

non parametric tests for right censored data with biased sampling testing the equality of two survival distributions can be difficult in a prevalent cohort study when non random sampling of subjects is involved due to the biased sampling scheme independent censoring assumption is often violated although the issues about biased inference caused by length biased sampling have been widely recognized in statistical epidemiological and economical literature there is no satisfactory solution for efficient two sample testing we propose an asymptotic most efficient nonparametric test by properly adjusting for length biased sampling the test statistic is derived from a full likelihood function and can be generalized from the two sample test to a k sample test the asymptotic properties of the test statistic under the null hypothesis are derived using its asymptotic independent and identically distributed representation we conduct extensive monte carlo simulations to evaluate the performance of the proposed test statistics and compare them with the conditional test and the standard logrank test for different biased sampling schemes and right censoring mechanisms for length biased data empirical studies demonstrated that the proposed test is substantially more powerful than the existing methods for general left truncated data the proposed test is robust still maintains accurate control of type i error rate and is also more powerful than the existing methods if the truncation patterns and right censoring patterns are the same between the groups we illustrate the methods using two real data examples

empirical likelihood for right censored lifetime data abstractwhen the empirical likelihood el of a parameter is constructed with right censored data literature shows that 2log empirical likelihood ratio typically has an asymptotic scaled chi squared distribution where the scale parameter is a function of some unknown asymptotic variances therefore the el construction of confidence intervals for requires an additional estimation of the scale parameter additional estimation would reduce the coverage accuracy for by using a special influence function as an estimating function we prove that under very general conditions 2log empirical likelihood ratio has an asymptotic standard chi squared distribution with one degree of freedom this eliminates the need for estimating the scale parameter as well as eases some of the often demanding computations of the el method our estimating function yields a smaller asymptotic variance than those of wang and jing 2001 and qin and zhao 2007 thus it is not surprising that confidence intervals usi

motivic serre invariants and weil restriction we study the interactions between weil restriction for formal schemes and rigid varieties greenberg schemes and motivic serre invariants and their behavior with respect to finite extensions of the base ring r which is a complete discrete valuation ring with perfect residue field

generating relations for reducing matrices i ordinary representations auxiliary groups are constructed that make it possible to reduce the multiplicity problem and to derive consistent generating relations for the elements of reducing matrices three examples are worked out to illustrate the general scheme

coefficients of fractional parentage for u m p n q u m n u p q and u m n u m u n the outer product reduction coefficients orc which reduce the representation rep induced from the irreps of the permutation groups s f1 and s f2 into the irreps of s f1 f2 are shown to be the indirect coupling coefficients for the u m p n q u m n u p q irreducible basis the non standard orc for reducing the rep induced from the non standard irreps of s f13 s f1 s f3 and s f24 s f2 s f4 into that of s f s f12 s f34 with fij fi fj f f12 f34 are identified with the u f u f12 u f34 clebsch gordan coefficient for the special gel fand bases of u f12 and u f34 the u m p n q u m n u p q cfp as well as its special case the u m p u m u p cfp are identified with the s f s f12 s f34 outer product isoscalar factor the u m n u m u n cfp are obtained from the u m n u m u n cfp by simply changing all the partition labels for u n into their conjugates interchanging rows with columns and taking into account a phase change the cfp can be calculated from the orc numerical values of the one body cfp for systems with up to six particles are tabulated

galois descent of semi affinoid spaces we study the galois descent of semi affinoid non archimedean analytic spaces these are the non archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring such as for example open or closed polydiscs or polyannuli using weil restrictions and galois fixed loci for semi affinoid spaces and their formal models we describe a formal model of a k analytic space x provided that x otimes kl is semi affinoid for some finite tamely ramified extension l of k as an application we study the forms of analytic annuli that are trivialized by a wide class of galois extensions that includes totally tamely ramified extensions in order to do so we first establish a weierstrass preparation result for analytic functions on annuli and use it to linearize finite order automorphisms of annuli finally we explain how from these results one can deduce a non archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero

symmetric and unitary group representations i duality theory an extension of the schur weyl duality connecting the representations of the symmetric and unitary groups is given the schur weyl basis is constructed using annihilation and creation operators three factorisation lemmas are derived their importance lies in the fact that they relate the phase freedoms within the racah wigner algebra of the symmetric groups and the unitary groups extensions of the regge symmetries are also given these are expressed in five duality relations

nonparametric estimation for right censored length biased data a pseudo partial likelihood approach to estimate the lifetime distribution of right censored length biased data we propose a pseudo partial likelihood approach that allows us to derive two nonparametric estimators with its closed form estimators and explicit limiting variances this approach retains the simplicity of conditional analysis and has only a small efficiency loss compared with the unconditional analysis under some regularity conditions we show that the two estimators are uniformly consistent and converge weakly to gaussian processes a simulation study demonstrates that the proposed estimators have satisfactory finite sample performance application to an alzheimer s disease study is reported

some 6 j symbols for symplectic and orthogonal groups by cerkaski s method the method devised by cerkaski 1987 has been used for the group sp 2n to find some 6 j symbols for which a single multiplicity index is required this approach possesses some advantages over that based on racah s techniques for so 3 the formulae for the 6 j symbols are comparatively simple and they automatically exhibit the symmetries of jucys 1973 elementary means are used to show that the eigenvalues of casimir s operator c2 for sp 2n are identical to those of c2 for o 2d if the authors take d n and replace the irreps of sp 2n by their transposed counterparts for o 2d where d is integral or half integral this enables the formulae for sp 2n to be used to generate expressions for some 6 j symbols for the orthogonal groups o 2d and so 2d illustrations are provided by so 2l 1 and o 4

generating relations for reducing matrices ii corepresentations the auxiliary group approach developed in paper 1 r dirl p kasperkovitz m i aroyo j n kotzev and m angelova tjurkedjieva j math phys 27 37 1986 is generalized for the case of corepresentations of antiunitary groups it allows us to reduce the multiplicity problem and to derive consistent generating relations for the elements of the reducing matrices for coreps two examples are worked out to illustrate the general scheme

a reduced subduction graph and higher multiplicity in sn transformation coefficients transformation coefficients between standard bases for irreducible representations of the symmetric group sn and split bases adapted to the subgroup n1 n2 n are considered we first provide a selection rule and an identity rule for the subduction coefficients which allow us to decrease the number of unknowns and equations arising from the linear method by pan and chen then using the reduced subduction graph approach we may look at higher multiplicity instances as a significant example an orthonormalized solution for the first multiplicity three case which occurs in the decomposition of the irreducible representation 4 3 2 1 of s10 into 3 2 1 3 1 of s6 s4 is presented and discussed

kernel density estimation for length biased data a new kernel density estimator for length biased data which derives from smoothing the nonparametric maximum likelihood estimator is proposed and investigated it has various advantages over an alternative method suggested by bhattacharyya franklin richardson 1988 it is necessarily a probability density it is particularly better behaved near zero it has better asymptotic mean integrated squared error properties and it is more readily extendable to related problems such as density derivative estimation

algebraic expressions for some generalised 6 j symbols for so n and g2 a class of generalised 6 j symbols which enter naturally when a second pair of modes epsilon tau 2 is added to the octahedral jahn teller system gamma 8 epsilon tau 2 has been evaluated for the group so 5 the method involves the fictitious boson configurations dwd and can be readily generalised to lwl the resulting 6 j symbols for so 2l 1 can be used to produce others by means of generalisations of the equations satisfied by the ordinary 6 j symbols for so 3 extensions to so 2l 2 and g2 are described as well as the weight space symmetries of jucys 1973 the relation between 6 j symbols and isoscalar factors is illustrated for so 7 contains implies g2

orientations of reflection rotation groups the authors extend the work of a previous paper 1980 in which they showed that some of the phase choices in the 3jm factors of certain point group embeddings affect the orientation of the symmetry axes the occurrence of such choices is unpredictable and their effect is sometimes subtle but once a set of 3jm factors has been calculated they can determine the orientations of the symmetry axes and mirror planes and investigate how these orientations vary with the phase choices the properties of the reflection rotation groups are obtained from the the isomorphic pure rotation groups but care is necessary in interpreting the effects of the corresponding operations of for example d3 and c3v

recoupling lie algebra and universal ω algebra we formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation this gives a generalization of graded lie algebra underlying this is the new notion of an algebra defined in this paper algebra is a generalization of algebra that goes beyond nonassociativity we construct the universal enveloping algebra of recoupling lie algebras and prove a generalized poincare birkhoff witt theorem as an example we consider the algebras over an arbitrary recoupling of zn graded heisenberg lie algebra finally we uncover the usual coalgebra structure of a universal envelope and substantiate its hopf structure

the symmetric group algebraic formulas for some sf 6j symbols and sf sf1 sf2 3jm symbols explicit rank dependent expressions have been obtained for some symmetric group sf 6j symbols and some sf sf1 sf2 3jm symbols using butler s recursion method a key point in deriving these results is the use of the reduced notation introduced by murnaghan to label irreps various symmetries of the 6j and 3jm symbols have been imposed these include the complex conjugation permutation and transpose conjugation we incorporate a new symmetry that arises from the occurrence of the two isomorphic direct product groups sf1 sf2 and sf2 sf1 as subgroups of sf in relation to the tables of 6j and 3jm symbols presented a discussion is given of the symmetric group unitary group duality

calculating symmetric group split standard transformation coefficients using the block selective method a proof the split basis for the symmetric group sn is adapted to sn and to the product subgroup sn a sa the entries in the matrix that transforms between the split basis representation and the standard young yamanouchi yy basis the split standard transformation coefficients are closely related to a particle coefficients of fractional parentage recently the block selective conjecture was proposed for calculating the coefficients the conjecture implies that the transformation matrix is directly related to the representation matrix in the yy basis of a cycle permutation we present a proof of this conjecture

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

symmetry properties of s classified su 3 3j 6j and 9j symbols starting from the s classified su 3 clebsch gordan coefficients introduced previously pluhar et al 1985 the s classified su 3 3j 6j and 9j symbols are constructed and examined they satisfy simple symmetry relations similar to those valid for the su 2 case

su mn su m su n isoscalar factors and s f1 f2 s f1 s f2 isoscalar factors a simple relation is found between the isoscalar factor isf of the unitary group and those of the permutation group i e the su mn su m su n isf is equal to the s f1 f2 s f1 s f2 isf since the values of s f1 f2 s f1 s f2 isf are independent of m and n one arrives at an important conclusion that the values of su mn su m su n isf are also independent of m and n therefore they can be calculated for all m and n by a single stroke instead of one m and one n at a time an eigenfunction metho for evaluating the su mn su m su n isf is given which can be easily translated into a computer program

symmetries of group subgroup transformation factors for coupling subduction and induction group subgroup gh transformation theory uses g transformation coefficients and gh transformation factors to describe the relationship between the various bases of representation space of any group decomposed into irreducible representation spaces of the group and its subgroups the representation spaces may be formed by the processes of coupling subduction or induction the authors further develop this gh transformation theory with the study of the various symmetries complex conjugation transposition and associativity of the gh transformation factor emphasising the origins of each symmetry and a hierarchy within these symmetries a method of calculation is given for certain transformation factors the results and methods presented here generalise those of the racah wigner coupling algebra

3jm and 6j tables for some bases of su6 and su3 tables of 6j symbols for su6 and su3 and tables of 3jm factors for su6 contains implies su2 su3 su3 contains implies u1 su2 and su3 contains implies so3 are presented the tables are computer produced using a program that implements the building up principle in a general form these tables are useful for calculations in high energy nuclear and solid state physics some other tabulations contain errors and none uses all the symmetries available the n independence of sun results is discussed by using the various symmetric group unitary group duality relations

coupling subduction and induction transformations for group representations the basis of a representation space of any group may be chosen according to its decomposition into representations of the group and a subgroup chain coupling recoupling subduction and resubduction factors are each shown to arise as a result of schur s lemmas and a particular group subgroup structure when induction and mackey s subgroup theorem are considered new transformation factors occur those of induction and reinduction key properties of these factors are given together with their relationship with double coset matrix elements

simultaneity and reality of u n and su n 3jm and 6j symbols a simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups u n and su n including the case of mixed tensor representations we observe that simultaneity is incompatible with the conventional permutation symmetries for u n with n 3 the relevance of these results to the schur weyl duality of u n with the symmetric groups sl is discussed

diagram projection rules for recoupling diagrams in the racah wigner category this paper gives three important techniques for recoupling coefficients of an arbitrary number or irreducible representations factorization as a product of smaller degree recoupling coefficients diagram projection projecting a recoupling diagram to obtain constraints on recoupling coefficients and primitive expansion expressing a recoupling coefficient in terms of primitive recoupling coefficients the first technique reduces a recoupling coefficient to its essential and simplest form the second is many faceted allowing one to work at the recoupling level importantly recoupling diagrams can be used to construct a complete set of constraints on the recoupling coefficients the third allows large degree recoupling coefficients to be expressed in terms of primitive recoupling coefficients an example of this is the fact that the 9j symbol is defined by the 3j and 6j symbols

algebraic expressions for some so n racah coefficients in this paper the algebraic expressions for so n racah coefficients involving the coupling of three irreducible representations 1 0 over dot m 1 m 2 m 3 0 over dot are given by using the building up process their properties are discussed in detail c 1998 american institute of physics s0022 2488 98 01811 8

racah coefficients of the lie superalgebra osp 1 2 using the indefinite metric the author studies the coupling of three irreps of the osp 1 2 algebra and gives the general definition orthogonality conditions symmetry properties and calculating formulae of the racah coefficients of the osp 1 2 algebra eight kinds of racah coefficients exist for the osp 1 2 algebra and they are all expressed in terms of the racah coefficients of the so 3 algebra

generating function of isoscalar factors of unitary groups the author introduces a new schematisation of the elements of the bases of the fundamental representations of the unitary groups using this schematisation as well as two rules which will be established in this work the author is able to find by an elementary calculation the generating function of coupling coefficients of this basis this last generating function permits the author to find the one corresponding to su n isoscalar factors the approach gives in the general case i e for every n the algebraic expressions of these factors the author shows that to find the elements of the irreducible representation of the direct product of n representations and gives the details of this method in the special case n 2 a priori the method can be easily applied to other classical groups

3jm factors and basis functions for d h and c v the results of earlier work by butler and reid for the pure rotation double group dinfinity are used to obtain algebraic formulae for the 3jm factors of the branching dinfinity h contains implies cinfinity v other branchings involving these groups are trivially related to branchings involving dinfinity the transformations of the group chains o3 contains implies dinfinity h contains implies cinfinity h and o3 contains implies dinfinity h contains implies cinfinity v contains implies cinfinity to the standard angular momentum basis are also given

estimation of incubation period and generation time based on observed length biased epidemic cohort with censoring for covid 19 outbreak in china the incubation period and generation time are key characteristics in the analysis of infectious diseases the commonly used contact tracing based estimation of incubation distribution is highly influenced by the individuals judgment on the possible date of exposure and might lead to significant errors on the other hand interval censoring based methods are able to utilize a much larger set of traveling data but may encounter biased sampling problems the distribution of generation time is usually approximated by observed serial intervals however it may result in a biased estimation of generation time especially when the disease is infectious during incubation in this paper the theory from renewal process is partially adopted by considering the incubation period as the interarrival time and the duration between departure from wuhan and onset of symptoms as the mixture of forward time and interarrival time with censored intervals in addition a consistent estimator for the distribution of generation time based on incubation period and serial interval is proposed for incubation infectious diseases a real case application to the current outbreak of covid 19 is implemented we find that the incubation period has a median of 8 50 days 95 confidence interval ci 7 22 9 15 the basic reproduction number in the early phase of covid 19 outbreak based on the proposed generation time estimation is estimated to be 2 96 95 ci 2 15 3 86

semiparametric model for bivariate survival data subject to biased sampling to better understand the relationship between patient characteristics and their residual survival after an intermediate event such as the local cancer recurrence it is of interest to identify patients with the intermediate event and then analyze their residual survival data one challenge in analyzing such data is that the observed residual survival times tend to be longer than those in the target population since patients who die before experiencing the intermediate event are excluded from the identified cohort we propose to jointly model the ordered bivariate survival data using a copula model and appropriately adjusting for the sampling bias we develop an estimating procedure to simultaneously estimate the parameters for the marginal survival functions and the association parameter in the copula model and use a two stage expectation maximization algorithm using empirical process theory we prove that the estimators have strong consistency and asymptotic normality we conduct simulations studies to evaluate the finite sample performance of the proposed method we apply the proposed method to two cohort studies to evaluate the association between patient characteristics and residual survival

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

maximum likelihood estimation under a successive sampling discovery model a common problem encountered in the analysis of discovery data is the size bias phenomenon in which the larger units tend to be discovered first one approach to account for this bias is to model the discovery process as sampling successively from a finite population without replacement and with probability proportional to size we consider in this article a generalized version of this model for analyzing multivariate data with any given measure of size we assume a superpopulation framework and develop procedures for maximum likelihood estimation of the parameters of the distribution the use of the em algorithm for computing the maximum likelihood estimates associated computational issues and relationships to regression estimators in survey sampling are discussed oil discovery data from the rimbey meadowbrook reef play are used to illustrate the techniques

the clebsch gordan coefficients of permutation groups s 2 s 6 the clebsch gordan cg coefficients of the permutation groups s 2 s 6 are calculated by using the eigenfunction method phases have been chosen in a consistent way to exhibit as many symmetries for the coefficients as is possible including non multiplicity free cases the cg coefficients of s 2 s 5 and those for 321 321 to 321 5 of s 6 are tabulated in the square root form of rationals these tables together with the s 6 contains implies s 5 isoscalar factors published earlier provide a complete tabulation of the s 2 s 6 cg coefficients given in the young yamanouchi basis

a class of generalised 9 j symbols for sp 2n the symbolic method of kramers 1930 is used to obtain an expression for a 9 j symbol in terms of multiple products of spinor invariants this technique is generalised from the unitary compact symplectic group sp 2 to sp 2n and a generating function is found for a class of multiplicity free 9 sigma symbols where sigma denotes an irreducible representation of sp 2n of the form sigma 1 sigma 2 sigma n for which sigma i 0 i 1 schwinger s generating function for a 9 j symbol is recovered by setting n 1 a specialisation to 6 sigma symbols is made by setting one of the nine irreducible representations equal to the scalar 00 0 and the method is checked by working out a sample 6 sigma symbol previously obtained by an aufbau approach

6 j for mixed symmetry triads in k20 the authors discuss the possible choices of conjugation and permutation matrices for mixed symmetry triads this requires a short review of the previous choices for all triads the primitive 6 j of the finite group k20 are then calculated as an example of a group that contains mixed symmetry triads

analyzing length biased data with semiparametric transformation and accelerated failure time models right censored time to event data are often observed from a cohort of prevalent cases that are subject to length biased sampling informative right censoring of data from the prevalent cohort within the population often makes it difficult to model risk factors on the unbiased failure times for the general population because the observed failure times are length biased in this paper we consider two classes of flexible semiparametric models the transformation models and the accelerated failure time models to assess covariate effects on the population failure times by modeling the length biased times we develop unbiased estimating equation approaches to obtain the consistent estimators of the regression coefficients large sample properties for the estimators are derived the methods are confirmed through simulations and illustrated by application to data from a study of a prevalent cohort of dementia patients

toute forme moderement ramifiee d un polydisque ouvert est triviale let k be a complete non archimedean field and let x be a k analytic space assume that there exists a finite tamely ramified extension l of k such that x l is isomorphic to an open polydisc over l we prove that x is itself isomorphic to an open polydisc over k the proof consists in using the graded reduction a notion which is due to temkin of the algebra of functions on x together with some graded counterparts of classical commutative algebra results nakayama s lemma going up theorem basic notions about etale algebras etc

on a labeling for point group harmonics ii icosahedral group the expressions for the 2 5 6 8 and 9 representations are given in terms of those of 1 representations for neighboring angular momenta the coefficients of the 4 representations are expressed in terms of those of 7 representations therefore with an arbitrary choice of orthonormal sets of 1 3 and 7 representations orthonormal sets of other kinds of representations are well defined and can be labeled with the labels of parent representations all clebsch gordan coefficients are expressed in terms of those between parent representations and a few others tables of all nondegenerate 1 3 and 7 representations are given for the axes of quantization of order 5 with some degenerate 3 and 7 representations which are also given any representation of integer or half integer angular momentum up to 27 can be obtained using some usual clebsch gordan coefficients of su 2

phase conventions consistent with duality between sun and sl the schur weyl construction of irreducible tensors by symmetrized lth rank exterior products of a defining n dimensional vector space establishes a duality between the coupling algebra of the unitary group un and the symmetric group sl the coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in un and association in sl the unitary group is factored by its unitary unimodular subgroup sun un u1 so it is usual to identify the coupling algebras of un and sun by choosing the trivial phaseless coupling for u1 this establishes a set of equivalence relations since all pseudoscalar irreducible representations irreps of un subduce onto the scalar irrep of sun the question remains can this trivial identification with the coupling algebra of sun be made consistent with the symmetries which follow from duality specifically the symmetries which follow from duality must be consistent with the equivalence relations we argue such consistency may be obtaine

a new duality theory for compact groups the tannaka krein duality theory characterizes the category g of finite dimensional continuous unitary representations of a compact group as a subcategory of the category of hilbert spaces we prove a more powerful result characterizing g as an abstract category every strict symmetric monoidalc category with conjugates which has subobjects and direct sums and for which thec algebra of endomorphisms of the monoidal unit reduces to the complex numbers is isomorphic to a category g for a compact groupg unique up to isomorphism

algebraic formulas for some nontrivial un 6j symbols and umn um un 3jm symbols we give tables of algebraic formulas for some nontrivial 6j symbols and 3jm symbols of the unitary groups the tables demonstrate that the building up method can be used successfully to obtain the rank dependence of unitary group j and jm symbols to emphasize the rank dependent nature of this calculation we have employed the composite young tableaux notation or back to back notation to label the unitary group irreps in using this notation the transpose conjugate symmetry of the corresponding composite young diagram leads to a new symmetry of the unitary group 6j and 3jm symbols the transposition of the groups um and un gives rise to a further symmetry of the 3jm symbols of umn um un

low risk population size estimates in the presence of capture heterogeneity summary population estimation methods are used for estimating the size of a population from samples of individuals in many applications the probability of being observed in the sample varies across individuals resulting in sampling bias we show that in this setting estimators of the population size have high and sometimes infinite risk leading to large uncertainty in the population size as an alternative we propose estimating the population of individuals with observation probability exceeding a small threshold we show that estimators of this quantity have lower risk than estimators of the total population size the proposed approach is shown empirically to result in large reductions in mean squared error in a common model for capture recapture population estimation with heterogeneous capture probabilities

algebraic expressions of the clebsch gordon coefficients of the point group t a method for finding algebraic expressions of the clebsch gordan cg coefficients of point groups is proposed and applied to the tetrahedral group it is shown that in constructing the cg coefficients the irreducible symmetry operator iso of a double group g can be replaced by an effective iso which is much simpler than the usual iso the effective iso for the group chain t c3 is p 3d c2z c2z where d c2z is the matrix of c2z in the irrep of t with this effective iso and the algebraic expression of d c2z the algebraic expressions are derived for the real cg coefficients of t in the group chain t c3 the algebraic expressions for the complex real cg coefficients of the group chain t d2 c2 t c2 have also been obtained

subduction coefficients of birman wenzl algebras and racah coefficients of the quantum groups oq n and spq 2m i subduction coefficients irreducible representations of birman wenzl algebras cf r q in the non standard basis are discussed a procedure for evaluating subduction coefficients sdcs or the transformation coefficients between standard and non standard basis of birman wenzl algebras is formulated based on the linear equation method sdcs of cf r q with f 4 are derived racah coefficients of the quantum groups oq n and spq 2m can be obtained from sdcs of birman wenzl algebras cf r q by using the schur weyl brauer duality relation between birman wenzl algebras cf r q and the quantum groups oq n and spq 2m

algebraic expressions for some multiplicity free 6j symbols and isoscalar factors for g2 generalized 6j symbols for g2 in which all four triangular conditions involve the seven dimensional irreducible representation irrep 10 are multiplicity free algebraic expressions depending on the highest weights of the accompanying irreps are found by using generalizations of the racah wigner algebra a rule is given for generalizing the so 3 phase factors the results are applied to finding algebraic expressions for a class of isoscalar factors for so 7 g2

empirical likelihood based inference in missing response problems and its application in observational studies summary the problem of missing response data is ubiquitous in medical and social science studies in the case of responses that are missing at random depending on some covariate information analyses focused only on the complete data may lead to biased results various debias methods have been extensively studied in the literature particularly the weighting method that was motivated by horvitz and thompson s estimators to improve efficiency robins rotnitzky and zhao proposed augmented estimating equations based on corrected complete case analyses a nice feature of the augmented method is its double robustness i e the estimator that is derived from the augmented method is asymptotically unbiased if either the underlying missing data mechanism or the underlying regression function is correctly specified furthermore the augmented estimator can achieve full efficiency if both the missing data mechanism and the regression function are correctly specified in general however it is very difficult to specify the regression function correctly especially when the dimension of covariates is high this is the so called curse of dimensionality problem the augmented estimator has much lower efficiency if the working regression model is not close to the true regression model in this paper the empirical likelihood method is employed to seek a constrained empirical likelihood estimation of mean response with the assumption that responses are missing at random the empirical likelihood based estimators enjoy the double robustness property moreover it is possible that the empirical likelihood based inference can produce asymptotically unbiased and efficient estimators even if the true regression function is not completely known simulation results indicate that the empirical likelihood based estimators are very robust to a misspecification of the propensity score and dominate other competitors in the sense of having smaller mean square errors methods that are developed in this paper have a nice application in observational causal inferences the propensity score is used to adjust for differences in pretreatment variables in the estimation of average treatment effects

merging information for semiparametric density estimation summary the density ratio model specifies that the likelihood ratio of m 1 probability density functions with respect to the mth is of known parametric form without reference to any parametric model we study the semiparametric inference problem that is related to the density ratio model by appealing to the methodology of empirical likelihood the combined data from all the samples leads to more efficient kernel density estimators for the unknown distributions we adopt variants of well established techniques to choose the smoothing parameter for the density estimators proposed

a duality consistent phase convention for complex conjugation in sun we have argued that a simple phase convention for all involutive mappings can be chosen so the algebra of inner product couplings in sun is identical to that for outer products in sl pure real phases with a single 1 entry in any row or column can represent the permutation matrix for transposition in order of the two component irreducible representations irreps in a binary coupling the matrix for association in sl with respect to the alternating group al the derome sharp matrix and the 1jm factor the latter two are required by complex conjugation in sun in our previous work we have proposed specific prescriptions for assigning the phase under transposition and association and have shown them to be consistent with duality in this work we propose a duality consistent prescription for assigning the phase of the derome sharp matrix and thus of the 1jm factor which is related to it by association