on resolving the multiplicity of arbitrary tensor products of the u n groups representations of u n are realised as right translations on holomorphic hilbert bargmann spaces of n n complex variables r fold tensor product spaces of irreducible representations of u n are shown to be isomorphic to subspaces of the holomorphic hilbert spaces maps are exhibited which carry an irreducible representation of u n into these subspaces the algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity several examples from u 3 are explicitly worked out

contrasting treatment specific survival using double robust estimators in settings where a randomized trial is infeasible observational data are frequently used to compare treatment specific survival the average causal effect ace can be used to make inference regarding treatment policies on patient populations and a valid ace estimator must account for imbalances with respect to treatment specific covariate distributions one method through which the ace on survival can be estimated involves appropriately averaging over cox regression based fitted survival functions a second available method balances the treatment specific covariate distributions through inverse probability of treatment weighting and then contrasts weighted nonparametric survival function estimators because both methods have their advantages and disadvantages we propose methods that essentially combine both estimators the proposed methods are double robust in the sense that they are consistent if at least one of the two working regression models i e logistic model for treatment and cox model for death hazard is correct the proposed methods involve estimating the ace with respect to restricted mean survival time defined as the area under the survival curve up to some prespecified time point we derive and evaluate asymptotic results through simulation we apply the proposed methods to estimate the ace of donation after cardiac death kidney transplantation with the use of data obtained from multiple centers in the netherlands

irreducible embeddings and polynomial tensors the problem of finding irreducible tensors which are polynomials in the components of an irreducible tensor n of a compact semisimple group h is simplified if the representation n is contained irreducibly in a representation m of a larger group g of which h is a subgroup a number of examples are given for which the problem is completely solved by generating function methods the authors discuss how one can use the computer to tackle the more difficult cases the problem of the reduction of the so 8 enveloping algebra is given as an example

testing violations of the exponential assumption in cancer clinical trials with survival endpoints personalized cancer therapy requires clinical trials with smaller sample sizes compared to trials involving unselected populations that have not been divided into biomarker subgroups the use of exponential survival modeling for survival endpoints has the potential of gaining 35 efficiency or saving 28 required sample size miller 1983 making personalized therapy trials more feasible however the use of exponential survival has not been fully accepted in cancer research practice due to uncertainty about whether or not the exponential assumption holds we propose a test for identifying violations of the exponential assumption using a reduced piecewise exponential approach compared with an alternative goodness of fit test which suffers from inflation of type i error rate under various censoring mechanisms the proposed test maintains the correct type i error rate we conduct power analysis using simulated data based on different types of cancer survival distribution in the seer registry database and demonstrate the implementation of this approach in existing cancer clinical trials

the polynomial type analysis of su 3 group theoretical quantities formulas for the transformation matrix between the canonical and noncanonical su 3 bases in both and cases are derived an expression for the isoscalar factor of the su 3 coupling coefficient in the noncanonical basis with one symmetric representation is also derived both expressions are put in a polynomial type form suitable for coding computer programs to obtain explicit and exact algebraic expressions

some special su 3 r 3 wigner coefficients and their application bargmann space expansions of oscillator functions are used to derive analytic expressions for su 3 r 3 wigner coefficients for the couplings 10 0 2 3 3 l3 0 and 10 20 3 3 l3 0 with arbitrary 3 3 these lead to expansions useful in nuclear cluster problems and are used to give a simple form for the su 3 irreducible tensor expansion of a scalar two body interaction an application which motivated this investigation

on the denominator function for canonical su 3 tensor operators the definition of a canonical unit su 3 tensor operator is given in terms of its characteristic null space as determined by group theoretic properties of the intertwining number this definition is shown to imply the canonical splitting conditions used in earlier work for the explicit and unique up to phases construction of all su 3 wcg coefficients wigner clebsch gordan using this construction an explicit su 3 invariant denominator function characterizing completely the canonically defined wcg coefficients is obtained it is shown that this denominator function squared is a product of linear factors which may be obtained explicitly from the characteristic null space times a ratio of polynomials these polynomials denoted gtq are defined over three shift parameters and three barycentric coordinates the properties of these polynomials hence of the corresponding invariant denominator function are developed in detail these include a derivation of their degree symmetries and zeros the

asymptotically fast factorization of integers the paper describes a probabilistic algorithm for finding a factor of any large composite integer n the required input is the integer n together with an auxiliary sequence of random numbers it is proved that the expected number of operations which will be required is o exp 83qn n in in n l 2 for some constant f 0 asymptotically this algorithm is much faster than any previously analyzed algorithm for factoring integers earlier algorithms have all required o na operations where a 1 5

decomposition of the so 8 enveloping algebra under u 4 u 3 the existence of a complete set of su 3 tensor operators in the enveloping algebra of so 8 is demonstrated the analysis recasts a parallel analysis by biedenharn and flath commun math phys 93 143 1984 concerning an so 6 2 model for su 3 in the isomorphic but simpler framework of an so 8 model

a solution to the su n external state labelling problem based upon a u n 1 n 1 group i general theory the state labelling problem arising in the reduction of the direct product of a p positive row u n irreducible representation h1 hp0 with a q negative row one 0 h q h 1 into a sum of mixed u n irreducible representations k1 kp0 k q k 1 is solved by using the complementarity between u n and u p q within some positive discrete series irreducible representations of u pn qn this complementarity allows analysis of the problem in terms of the group chain u p q contains implies u p u q instead of u n u n contains implies u n for the most general su n irreducible representations corresponding to p q n 1 the relevant group chain is therefore u n 1 n 1 contains implies u n 1 u n 1 in such a case the additional labels include those of an intermediate u n 1 irreducible representation h1s hn 1s as well as the additional labels solving the state labelling problems for the products k1 kn 1 h1s hn 1s and k 1 k n 1 h1s hn 1s of u n 1 irreducible representations hence the proposed solution reflects in a direct way the operation of king s branching rule for the chain u n u n contains implies u n supplemented whenever necessary with king s modification rule

new realizations of the dual space of su 3 the logic of constructing and interrelating classical realizations of the dual space of so 3 is exhibited analogies with different methods of quantization of gauge theories are pointed out the analysis is applied to su 3 to obtain existing realizations and to construct new ones the gelfand zetlyn basis for the irreducible representations of su 3 is explicitly realized using polynomials in four variables and positive or negative integral powers of a fifth variable another realization uses a spinor of so 6 so 3 1 these are the analogues of schwinger bargmann construction for su 2

coherent states for the harmonic series irreducible representations of u p q i unitary operator coherent states unitary operator coherent states as defined by klauder 1963 perelomov 1972 and gilmore 1974 are considered for the harmonic series irreducible representations of u p q various properties of these states are reviewed including their reproducing kernel and measure explicit forms as well as the representation they lead to for the u p q generators starting from a contraction of the su p q lie algebra both dyson and holstein primakoff boson realisations of u p q are obtained in terms of pq pairs of boson creation annihilation operators and of the generators of a u p u q intrinsic group such boson realisations are applied to determine the matrix elements of the u p q generators between discrete bases classified according to the chain u p q contains implies u p u q finally the isomorphism between so 4 2 and su 2 2 is employed to derive the corresponding properties of the so 4 2 unitary operator coherent states for application in atomic physics

on resolving the multiplicity of tensor products of irreducible representations of symplectic groups the multiplicity problem of r fold tensor products of irreducible representations of sp 2k c are considered the arbitrary r fold tensor product is shown to be isomorphic to a subspace of the holomorphic hilbert bargmann space of n 2k complex variables maps are constructed which carry an irreducible representation of sp 2k c into this subspace an algebra of commuting operators is constructed and it is shown how eigenvalues and eigenvectors of certain of these operators can be used to resolve the multiplicity

holomorphic discrete series and harmonic series unitary irreducible representations of non compact lie groups sp 2n r u p q and so 2n generalised characters of the infinite dimensional holomorphic discrete series unitary irreducible representations of the non compact groups u p q sp 2n r and so 2n are explicitly expressed in terms of characters of finite dimensional unitary group representations these formulae are remarkably succinct despite involving certain infinite series of schur functions similar formulae are derived for harmonic series unitary representation of both u p q and sp 2n r consideration of the branching rules from u p q to u q p and from sp 2n r to u n enables holomorphic representations to be identified as a subset of the harmonic representations the branching rules are established in full generality and are then used in the evaluation of tensor products of both holomorphic and harmonic representations in the case of the former a known result is recast in terms of closed formulae involving schur functions and for the latter various generalisations of these formulae are given a conjecture is also made regarding what might be the simplest possible formulae covering all holomorphic and harmonic representations of sp 2n r and u p q illustrative examples are presented

what price kaplan meier summary the asymptotic efficiency of the kaplan meier product limit estimator relative to the maximum likelihood estimator of a parametric survival function is examined under a random censoring model

canonical solution of the state labelling problem for su n so n and littlewood s branching rule i general formulation the internal state labelling problem for the d row irreducible representations of su n where 2d or n when reduced with respect to so n is shown to amount to the external state labelling problem for u d the canonical solution of the latter due to biedenharn et al 1967 provides a canonical solution of the former which reflects the operation of littlewood s branching rule for u n contains implies o n in a very simple way

restricted mean life with adjustment for covariates abstract although the mean is the most widely used measure of location in comparing two distributions its use is essentially precluded in the presence of substantial censoring typical in survival analysis of medical therapies to accommodate censoring irwin 1949 proposed estimation of the expectation of life restricted to a time t using the actuarial estimator of the survival function in this article we examine the restricted mean as an index for comparing survival in two groups and then extend the method to incorporate covariates into the analysis for this purpose it is convenient to depart slightly from the actuarial approach and to consider piecewise exponential models asymptotic properties are explored and the assumption of piecewise exponentiality is shown to be rather mild as departures from the model have a relatively small effect on estimation of the restricted mean the methodology is illustrated using published data from a study conducted by the radiation therapy oncology group rtog

canonical solution of the state labelling problem for su n so n and littlewood s branching rule iii su 3 so 3 case for pt ii see ibid vol 17 no 4 p 777 89 1984 parts i and ii discussed in general terms a new solution to the state labelling problem for the d row irreducible representations irreps of su n when reduced with respect to so n this solution was termed canonical because it reflects the operation of littlewood s branching rule for u n o n in a straightforward way the su 3 so 3 case is now worked out in detail explicit expressions of the canonical basis states of so 3 irreps l belonging to an su 3 irrep h1h2 are obtained the matrix of the transformation from the bargmann moshinsky basis to the canonical one is also calculated it is shown that in both cases the extra label necessary to completely specify the states can be chosen as the label js characterising an intermediate su 2 irrep in the reduction of the produce representation jl js into su 2 irreps j where j 1 2 h1 h2 and jl 1 2l or 1 2 l 1 whenever h1 h2 l is even or odd

the price of kaplan meier miller has studied the asymptotic efficiency of the nonparametric kaplan meier survival estimator relative to parametric estimates based on the exponential and weibull distributions he concluded that in certain cases the asymptotic efficiency is low and recommended that analysts give more consideration to parametric estimators particularly for estimation of small tail probabilities in this article we revisit this issue and examine the performance of the nonparametric procedure for estimation not only of a point on the survival curve but also of the mean or restricted mean lifetime in addition to the exponential and weibull families we consider the performance of the kaplan meier procedure relative to a more flexible parametric model proposed by efron we find that the reduction in efficiency of the kaplan meier survival estimate becomes negligible fairly quickly as the number of parameters in the parametric model increases moreover for estimation of the mean or restricted mean the loss in eff

the so 6 2 model of su 3 and its generalisation to su n the complementarity relation between the u p q and u n lie groups enables the authors to reformulate the so 6 2 model of su 3 in terms of a u 1 1 group and to outline its generalisation to a family of n 2 models of su n for n or 3 respectively associated with u n 2 1 u n 1 2 and u 1 n 2 groups

a solution to the su n external state labelling problem based upon a u n 1 n 1 group ii recursion relations for the su 2 and su 3 wigner coefficients for pt i see ibid vol 20 p 769 84 1987 the new solution to the su n external state labelling problem proposed in the first paper of this series is analysed in detail in the su 3 case in terms of the two complementary chains u 3 u 3 contains implies u 3 and u 2 2 contains implies u 2 u 2 the classification of su 3 coupled states is completed by the labels of an intermediate u 2 irreducible representation hs hs1hs2 directly related to king s branching rule for u 3 u 3 u 3 for pedagogical purposes the su 2 coupled state construction based upon the complementary chains u 2 u 2 contains implies u 2 and u 1 1 contains implies u 1 u 1 is considered first for both su 2 and su 3 it is proved that in addition to the standard recursion relations the corresponding wigner coefficients satisfy recursion relations of a different type arising from the action of the u 1 1 or u 2 2 complementary group generators a detailed example shows that such complementary recursion relations are quite useful for numerical purposes

the theory of angular momenta and higher su n symmetries the authors discuss interrelations between generating invariants gi for the reduction pi mi 1 d pi to d on 1 of the m fold tensor product of su n irreducible representations irreps d pi and polynominal bases of the su m irreps d on 1j om n 1 ok 0 0 0 k times a realisation of the su m irrep d oj om 3 bases is given in terms of gi for su 2 coupling wigner coefficients as a byproduct an expression is obtained for su 2 6 j symbols in terms of only two wigner coefficients the authors also discuss some possibilities of the analysis involved in solving the wigner biedenharn problem construction of orthonormal sets of the wigner coefficients for su n groups n or 3

vector coherent state theory of the generic representations of so 5 in an so 3 basis for applications of group theory in quantum mechanics one generally needs explicit matrix representations of the spectrum generating algebras that arise in bases that reduce the symmetry group of some hamiltonian of interest here we use vector coherent state techniques to develop an algorithm for constructing the matrices for arbitrary finite dimensional irreps of the so 5 lie algebra in an so 3 basis the so 3 subgroup of so 5 is defined by regarding so 5 as linear transformations of the five dimensional space of an so 3 irrep of angular momentum two a need for such irreps arises in the nuclear collective model of quadrupole vibrations and rotations the algorithm has been implemented in maple and some tables of results are presented

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

a calculus for su 3 leading to an algebraic formula for the clebsch gordan coefficients we develop a simple computational tool for su 3 analogous to bargmann s calculus for su 2 crucial new inputs are i explicit representation of the gelfand zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable ii an auxiliary gaussian measure with respect to which the gelfand zetlin states are orthogonal but not normalized iii simple generating functions for generating all basis states and also all invariants as an illustration of our techniques an algebraic formula for the clebsch gordan coefficients is obtained for the first time this involves only gaussian integrations thus su 3 is made as accessible for computations as su 2 is

canonical orthonormal basis for su 3 so 3 i construction of the basis a canonical orthonormal basis is given for generic representations of the group chain su 3 contains implies so 3

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

the remarkable algebra so 2n its representations its clifford algebra and potential applications properties of the real lie algebra so 2n and its finite dimensional representations are described the structure of the clifford algebras associated with the two fundamental spinor representations are determined and it is found in particular that for so 4n there is a remarkable asymmetry one representation is real the other complex though equivalent to its complex conjugate and the associated clifford algebras are inequivalent generalised dirac matrices are defined acting within the direct sum of these two spinor representations of so 2n these matrices generate a different clifford algebra not determined uniquely by those representations and their commutators do not span a representation of so 2n the isomorphisms of so 6 with su 3 1 and of so 8 with so 6 2 are described together with the corresponding mappings between finite dimensional irreducible representations a possible application to unified field theory is discussed

unitary representations branching rules and matrix elements for the non compact symplectic groups the complementarity of the symplectic and orthogonal groups is used to infer properties of the infinite dimensional unirreps of the former from the character theory of the latter the complete set of d series metaplectic unirreps of sp n r is identified and branching rules are given for their restrictions to the maximal compact subgroup u n developed in terms of the properties of schur functions a known algorithm for the evaluation of matrix elements of the sp 3 r lie algebra is extended to any sp n r and analytic expressions are given for important classes of unirreps and multiplicity free states

univariate polynomial factorization over finite fields abstract this paper is a tutorial introduction to univariate polynomial factorization over finite fields we recall the classical methods that induced most factorization algorithms berlekamp s and the cantor zassenhaus ones and some refinements which can be applied to these methods explicit algorithms are presented in a form suitable for almost immediate implementation we give a detailed description of an efficient implementation of the cantor zassenhaus algorithm used in the release 2 of the axiom computer algebra system

canonical orthonormal basis for su 3 contains implies so 3 iii complete set of su 3 tensor operators for pt ii see ibid vol 18 p 1905 1985 a complete set of tensor operators for su 3 is given in a model space generated by two bargmann vectors use is made of modified operator patterns to classify these tensors their tensorial properties are then discussed in the context of the su 3 tensor algebra developed by biedenharn and collaborators 1957 the authors also given analytical semi analytical expressions for important classes of wigner coefficients in gel fand rotational basis

logistic regression survival analysis and the kaplan meier curve abstract we discuss the use of standard logistic regression techniques to estimate hazard rates and survival curves from censored data these techniques allow the statistician to use parametric regression modeling on censored data in a flexible way that provides both estimates and standard errors an example is given that demonstrates the increased structure that can be seen in a parametric analysis as compared with the nonparametric kaplan meier survival curves in fact the logistic regression estimates are closely related to kaplan meier curves and approach the kaplan meier estimate as the number of parameters grows large

determination of the weyl invariants and calculation of the coupling coefficients for unitary groups the author presents a new approach for determining the weyl invariants of unitary groups such an approach is based on the generating function of the basis of the representation for unitary groups built in an earlier work hage hassan 1983 the author obtains two expressions for the weyl invariants both including the parameters involved in the generating function coupling coefficients appear in one of the expressions the comparison of the development of the two expressions allows the calculation of the coupling coefficients the application of this approach to the calculation of the coupling coefficients for the direct product lambda 10 x lambda 2 mu 2 of su 3 leads to an expression of the isoscalar factor where no summation occurs in contradistinction with the known expression for the coupling coefficients which involve five summations

su n irreducible schwinger bosons we construct su n irreducible schwinger bosons satisfying certain u n 1 constraints which implement the symmetries of su n young tableaues as a result all su n irreducible representations are simple monomials of n 1 types of su n irreducible schwinger bosons further we show that these representations are free of multiplicity problems thus all su n representations are made as simple as su 2

complementary group resolution of the su n outer multiplicity problem ii recoupling approach for su 3 u 2 reduced wigner coefficients a general procedure for the derivation of su 3 u 2 reduced wigner coefficients rwcs for the coupling 1 1 2 2 where is the outer multiplicity label required in the decomposition is proposed based on a recoupling approach that follows the complementary group technique for a resolution of the outer multiplicity of su n introduced in part i of this series rwcs of su n are not unique under a canonical resolution of the outer multiplicity the transformation from one set to another are elements of so m where m is the number of occurrences of the irrep in the decomposition 1 1 2 2 a special resolution of the multiplicity is identified that leads to a recursive procedure for the determination of rwcs new features of these special rwcs and differences from those obtained with other choices are discussed the method can be applied to the derivation of general su n wigner or rwcs algebraic expressions for another kind of rwcs the so called reduced auxiliary wigner coef

models for su 3 in terms of so n 2 and so 2n algebras mathematical and physical models for su 3 are given in terms of unirreps of the so n 2 and so 2n algebras vector coherent state representations are given for all the discrete series unirreps of so n 2 and so 2n traceless bosons are shown to arise naturally in these representations a new and elegant realisation of the so 8 approximately so 6 2 model for su 3 is given and it is shown how to perform calculations in the so 8 model using the powerful vector coherent state representations formalism

a new boson calculus for su 3 it is shown that the gelfand weyl pattern for su 3 can be uniquely expressed in terms of four non negative and one positive or negative free integers this provides an optimal albeit non linear boson calculus for su 3 in terms of four harmonic oscillators and one planar rotor by optimal it is meant that every irreducible representation is obtained once and once only our techniques can be generalized to other groups

a simplified so 6 2 model of su 3 a new realization is obtained of the representation of so 6 2 which has been shown recently by flath and biedenharn and also by bracken and macgibbon to define a model of su 3 in contrast to the realization in terms of six pairs of boson operators used previously which involved cubic expressions the new realization involves only quadratic expressions in eight pairs of boson operators and is manifestly hermitian properties of this new oscillator realization and in particular its advantages over the old realization are discussed briefly it is deduced that the representation of so 6 2 is integrable to a unitary group representation

on the structure of tensor operators in su3 a global algebraic formulation of su3 tensor operator structure is achieved a single irreducible unitary representation irrep v ofso 6 2 is constructed which contains every su3 irrep precisely once an algebra of polynomial differential operatorsa acting onv is given the algebraa is shown to consist of linear combinations of all su3 tensor operators with polynomial invariant operators as coefficients by carrying out an analysis ofa the multiplicity problem for su3 tensor operators is resolved

creation and annihilation operators for su 3 in an so 6 2 model creation and annihilation operators are defined which are wigner operators tensor shift operators for su 3 while the annihilation operators are simply boson operators the creation operators are cubic polynomials in boson operators together they generate under commutation the lie algebra of so 6 2 the vector space generated from a vacuum vector by repeated application of the creation operators carries an irreducible representation of the so 6 2 algebra equivalent to an hermitian representation and also carries in direct sum every different irreducible representation of su 3 so 6 2 exactly once a model for su 3 in the sense of bernstein gel fand and gel fand is therefore defined the different su 3 irreducible representations appear explicitly as manifestly covariant irreducible tensors whose orthogonality and normalisation properties are examined other wigner operators for su 3 can be constructed simply as products of the new creation and annihilation operators or sums of such products

second degree kinematical constraints associated with dynamical symmetries the tensor operators that transform under subrepresentations of the symmetrized kronecker square of the adjoint representation for the classical semisimple lie algebras have been determined all irreducible representations that satisfy the identities obtained by the vanishing of these tensor operators have been deduced the pairs of irreducible representations of nonexceptional semisimple lie algebras the kronecker product of which decomposes into two irreducible components have been pointed out as a consequence of this result a comparison of the identities derived in the way indicated before with those resulting by the application of a method due to hannabuss kostant and okubo has been obtained connections between these results and those obtained by drinfeld concerning the solutions of quantum yang baxter equations have been established

complementary group resolution of the su n outer multiplicity problem i the littlewood rules and a complementary u 2n 2 group structure a complementary group to su n is found that realizes all features of the littlewood rules for kronecker products of su n representations this is accomplished by considering a state of su n to be a special gel fand state of the complementary group u 2n 2 with labels of the latter used to distinguish multiple occurrences of irreducible representations of su n irreps in the su n su n su n decomposition that is obtained from the littlewood rules furthermore this realization also helps us to determine su n su n 1 u 1 reduced wigner coefficients rwcs frequently called isoscalar factors and clebsch gordan coefficients cgcs or full nonreduced wigner coefficients of su n using algebraic or numeric methods in either the canonical or a noncanonical basis new explicit formulas for the su 3 and su 4 multiplicities are obtained by using this technique

a closed formula for the product of irreducible representations of su 3 we determine a closed formula in terms of p q r and s for the decomposition of the product p q r s of finite dimensional irreducible representations of su 3 we also determine in terms of p q r s m and n necessary and sufficient conditions that a term m n appears in this decomposition and its multiplicity

canonical solution of the state labelling problem for su n so n and littlewood s branching rule ii use of modification rules for pt i see ibid vol 16 p 2095 1983 in part i it was shown that when d 1 2n case where littlewood s branching rule 1950 for u n contains implies o n has to be supplemented with newell s modification rules 1951 the explicit solution of the state labelling problem for v 1 row irreducible representations of both su 2v 1 and su 2v is given and is shown to reflect the operation of littlewood s modified branching rule in a direct way the generalisation to higher d values is then outlined