generalized vector coherent state theory of sp 2n r vector operators and of sp 2n r contains implies u n reduced wigner coefficients sp 2n r contains implies u n reduced wigner coefficients coupling a positive discrete series unitary irreducible representation with the 2n dimensional non unitary irreducible representation characterized by 1 are evaluated by making use of generalized vector coherent state techniques general formulae are given in terms of the k matrix elements of vector coherent state theory and of some u n 6j or 9j type recoupling coefficients explicit analytic expressions are obtained in some important special cases such results should allow a determination of wsp 2n r approximately w n p 2n r and osp p 2n r matrix representations to be completed because both the w n generators and the osp p 2n r odd generators transform under the sp 2n r irrep 1

a differential operator realisation approach for constructing casimir operators of non semisimple lie algebras we introduce a search algorithm that utilises differential operator realisations to find polynomial casimir operators of lie algebras to demonstrate the algorithm we look at two classes of examples 1 the model filiform lie algebras and 2 the schrodinger lie algebras we find that an abstract form of dimensional analysis assists us in our algorithm and greatly reduces the complexity of the problem

analytic expressions for the matrix elements of generators of sp 6 in an sp 6 u 3 basis work done by many authors indicates that important tools for calculations in microscopic collective models are the matrix elements of the generators of the symplectic group sp 6 in an sp 6 u 3 basis rosensteel has derived recursion relations for these matrix elements while filippov has determined them using generating function technique but it would also be convenient to have explicit and analytic formulas for them this is what we do in this paper for the case of closed shells i e when the irreducible representation irrep of sp 6 is characterized by equal values for the three weight generators in the lowest weight state we also indicate how our results can be extended to the case of arbitrary irreps of sp 6 i e when we have open shells

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

matrix representation of the generators of symplectic algebras ii the general case with explicit results for sp 6 r pt i not available one of the important problems in lie algebras is to determine the matrix representation of their generators in a basis associated with a given irrep of the corresponding group for the unitary and orthogonal groups these representations were obtained by gelfand and zetlin in the present paper the authors give results of the same degree of generality for the symplectic lie algebras sp 2d r where d is any integer when the irreps are in the positive discrete series the basis is constructed from powers of the raising generators acting on gelfand states using the dyson boson realisation of the generators of sp 2d r and the result of gelfand for unitary lie algebras the matrix representation of these generators was obtained in closed form explicit results are given for sp 6 r and the authors sketch their application to the symplectic model of the nucleus

a new holstein primakoff realisation of sp 4 r a new holstein primakoff type realisation of the non compact lie algebra sp 4 r is given in a completely explicit analytic and closed form the construction is related to the representations where the casimir invariants of the subalgebras sp1 2 2 r are diagonal rather than the invariants of the maximal compact subalgebra u 2

optimum sample size determination in stratified case control studies with cost considerations we investigate sample size determination for cochran s test for stratified case control studies when samples of cases and controls are allocated to maximize the asymptotic efficiency of cochran s test subject to fixed total cost with cost per control varying by strata we consider two situations typical of strata matched case control studies when one samples both cases and controls and when cases are given and one samples controls in each situation we develop and study an asymptotic method for finding the sample size required for a specific power under the optimum allocation proposed by nam and fears also for the second situation we investigate an asymptotic method for determining the common ratio k in one to k strata matched case control studies without cost consideration for a given power when cases are given neither the optimum nor the standard control sample sizes appear in a closed form we present numerical methods for calculating these sample sizes and illustrate them with examples we find the reduction in total cost obtained under the optimum allocation compared to standard allocation more pronounced as the differences in stratum specific costs of sampling controls increase

coherent state theory of the noncompact symplectic group an extended coherent state theory is presented for the noncompact sp 3 r group which reveals a simple relationship between the sp 3 r algebra and its contracted u 3 boson limit the relationship is used to derive a remarkably accurate analytic expression for sp 3 r matrix elements for the generic lowest weight representations the expression is shown to be exact whenever the states involved are multiplicity free with respect to the u 3 subalgebra it is further shown how exact matrix elements are easily calculated in general dyson and holstein primakoff type u 3 boson expansions are given

optimum allocation of samples in strata matching case control studies when cost per sample differs from stratum to stratum we investigate the allocation of samples over strata in case control studies with consideration of sampling costs we consider two situations typical of strata matched case control studies subject to a fixed total cost we derived optimum allocations that maximize the asymptotic efficiency of cochran s test that is minimize the variance of the maximum likelihood estimator of the common odds ratio when the common odds ratio is one our results show that the standard design with equal numbers of cases and controls in each stratum is inefficient when costs per control differ greatly from stratum to stratum we show that the optimum design suggested in this paper is robust for values of the odds ratio common in epidemiologic studies

boson realization of sp 4 i the matrix formulation holstein and primakoff derived long ago the boson realization of a su 2 lie algebra for an arbitrary irreducible representation irrep of the su 2 group the corresponding result for su 1 1 sp 2 is also well known this raises the question of whether it is possible to obtain in an explicit analytic and closed form and for any integer d the boson realization of a sp 2d lie algebra for an arbitrary irrep of the sp 2d group which is a problem of considerable physical interest the case d 2 already illustrates the problem in its full generality and thus in this paper we concentrate on sp 4 the dyson realization is well known and the passage to bosons satisfying the appropriate hermiticity conditions can be done by a similarity transformation through an operator k what we want though is an explicit boson realization for sp 2d similar to the one that exists for sp 2 in sec vi we show how we can get it for sp 4 if the operator k is known unfortunately while the matrix form of k2 can be ex

dynamical group of microscopic collective states iii coherent state representations in d dimensions the present series of papers deals with various realizations of the dynamical group spc 2d r of microscopic collective states for an a nucleon system in d 1 2 or 3 dimensions when these collective states are assumed to be invariant under the orthogonal group o n associated with the n a 1 relative jacobi vectors in this paper we further study the barut girardello representation proposed in the first two papers of the present series to show that it may be reformulated in terms of some coherent states by generalizing to spc 2d r a class of sp 2 r coherent states introduced by barut and girardello for such purpose our starting point is another coherent state representation namely the perelomov one previously considered by kramer we also propose a third new class of coherent states leading to holstein primakoff representation in a straightforward way we review various properties of these three classes of coherent states such as their reproducing kernel and measure explicit forms their gener

closed analytic boson realizations for sp 4 the problem of determing a boson realization for an arbitrary irrep of the unitary simplectic algebra sp 2d or of the corresponding discrete unitary irreps of the unbounded algebra sp 2d r has been solved completely in recent papers by deenen and quesne j deenen and c quesne j math phys 23 878 2004 1982 25 1638 1984 26 2705 1985 and by moshinsky and co workers o castanos e chacon m moshinsky and c quesne j math phys 26 2107 1985 m moshinsky boson realization of symplectic algebras to be published this solution is not known in closed analytic form except for d 1 and for special classes of irreps for d 1 a different method of obtaining a boson realization that solves the full problem for sp 4 is described the method utilizes the chain sp 2d su 2 su 2 su 2 d times which for d 4 does not provide a complete set of quantum numbers though a simple solution of the missing label problem can be given this solution does not help in the construction

boson realisations of lie algebras and expansion of shift operators a shift operator expansion is used to extend the range of the boson realisation method of calculation of lie algebra generator matrix elements to those chains of algebras for which the original technique cannot be applied the example of wsp 4 r w 2 sp 4 r contains implies sp 4 r is treated in detail and a generalisation to wsp 2d r w d contains implies sp 2d r sp 2d r is pointed out

memoryless stabilization of uncertain dynamic delay systems riccati equation approach the authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems the uncertainties are time varying and within a given compact set this method is an extension of the riccati equation approach proposed by i r petersen and c v hollot 1986 the extension is straightforward also the uncertainties do not need to satisfy the matching conditions

vector coherent state theory as a theory of induced representations a general definition is given of vector coherent state vcs representation theory it is shown that the theory is more general than suggested by previous applications and that it incorporates the standard theories of induced representations as special cases the associated k matrix theory is also given a fuller treatment than hitherto and shown to provide a rather general algorithm both for projecting vcs representations from larger representations in which they are embedded and for determining the hermitian form with respect to which an isometric equivalent representation is in fact isometric

brief generalized popov theory applied to state delayed systems this paper deals with the generalized popov theory applied to linear delay systems sufficient conditions for memoryless stabilization as well as for coerciveness of an appropriate quadratic cost are given in terms of algebraic properties of some matrix pencil

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

intrinsic formulae for the casimir operators of semidirect products of the exceptional lie algebra g2 and a heisenberg lie algebra we show that the casimir operators of the semidirect products of the exceptional lie algebra g2 and a heisenberg algebra can be constructed explicitly from the casimir operators of g2

control sample size when cases are given in constant ratio stratum matched case control studies strata matched case control studies based on a given number of cases and k times as many controls are common we obtain a necessary and sufficient condition for there to exist a numerical solution for k with a desired level of a power we derive the maximum power of the cochran test which may be less than one when the number of cases is fixed we provide an approximate formula for the minimum number of cases required for a specified power there exists a numerical solution that is convergence of an iterative method for k if the number of cases given is greater than this minimum number and no such a solution otherwise we also show that the incremental gain in relative efficiency of the test with respect to k is diminished as k gets large

generalised holstein primakoff theory for anharmonic lattices the spectrum of elementary excitations for scalar field theories on a lattice is calculated in the large n approximation and is found to consist of two distinct branches one of which may be interpreted as a collective mode the method of calculation is based on a generalised holstein primakoff theory for the real symplectic algebra

vector coherent states of non compact orthosymplectic lie supergroups vector coherent states are defined for the generic lowest weight ladder irreducible representations of the non compact orthosymplectic lie supergroups osp 2m 2n r and osp 2m 1 2n r their simultaneous use with k matrix theory is shown to provide a powerful tool for the study of such representations this is illustrated with the detailed example of osp 1 2n r

sample size for case control studies using cochran s statistic sample size determination for case control studies of chronic disease are often based on the simple 2 x 2 tabular cross classification of exposure and disease thereby ignoring stratification which may be considered in the analysis one consequence of this approach is that the sample size may be inadequate to attain a specified power and size when performing a statistical analysis on j 2 x 2 tables using cochran s 1954 biometrics 10 417 451 statistic or the mantel haenszel 1959 journal of the national cancer institute 22 719 748 statistic a sample size formula is derived from cochran s statistic and it is compared with the corresponding one derived when the data are treated as unstratified and also with two other formulas proposed for stratified data analysis the formula developed yields values slightly higher than one recently proposed by munoz and rosner 1984 biometrics 40 995 1004 which assumes that both margins of each 2 x 2 table are fixed while the present study considers only the case control margin to be fixed

vector coherent state theory in a group with non commuting raising generators vector coherent state theory is used to give an explicit construction of the irreducible representations of a group with non commuting raising generators the simple example of the compact unitary symplectic group usp 4 contains implies usp 2 usp 2

dynamical group of microscopic collective states i one dimensional case in the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an a nucleon system defined as those a particle states invariant under the orthogonal group o n associated with the n a 1 jacobi vectors the present paper discusses the case of a hypothetical one dimensional space simple invariance considerations show that the dynamical group of collective states is then the group spc 2 r which is the restriction to the collective subspace of the group sp 2 r of linear canonical transformations in n dimensions conserving the o n symmetry in addition to the well known realization of the dynamical group in the schrodinger representation based upon the dzublik zickendraht transformation two new realizations are proposed the first acts in a barut hilbert space which is the subspace of a bargmann hilbert space of analytic functions left invariant by o n a unitary mapping is established between the ordi

collective nuclear potential within the microscopic sp 6 r model an algorithm for projecting a microscopic nuclear hamiltonian with a central interaction onto the enveloping algebra of the sp 6 r group is presented in parallel with the development of a practical procedure for constructing the corresponding part of the sp 6 r so 3 integrity basis the latter is shown to consist of the invariants of the monopole quadrupole tensor and its polynomial analogs coupling the collective and vortex spin degrees of freedom the algorithm uses vector coherent state techniques and is displayed on the basis of a generic sp 6 r irreducible representation the collective hamiltonian is also shown to be expandable in terms of the enveloping algebra of the a 1 dimensional rotational group where a is the total number of nucleons

vector coherent state theory of the semidirect sum lie algebras wsp 2n r positive discrete series irreducible representations of the semidirect sum lie algebras wsp 2n r w n sp 2n r are given in wsp 2n r contains implies sp 2n r contains implies u n bases the branching rule for the reduction of a wsp 2n r irreducible representation into a direct sum of multiplicity free sp 2n r irreducible representations is accounted for by the explicit construction of a wsp 2n r contains implies sp 2n r contains implies u n basis of vector valued holomorphic functions

boson representations of the real symplectic group and their applications to the nuclear collective model both non hermitian dyson and hermitian holstein primakoff representations of the sp 2d r algebra are obtained when the latter is restricted to a positive discrete series irreducible representation d n 2 1 n 2 for such purposes some results for boson representations recently deduced from a study of the sp 2d r partially coherent states are combined with some standard techniques of boson expansion theories the introduction of usui operators enables the establishment of useful relations between the various boson representations two dyson representations of the sp 2d r algebra are obtained in compact form in terms of d d 1 2 pairs of boson creation and annihilation operators and of an extra u d spin characterized by the irreducible representation 1 d in contrast to what happens when 1 d it is shown that the holstein primakoff representation of the sp 2d r algebra cannot be written in such a compact form for a generic irreducible representation explicit expansions are

new coherent states and inhomogeneous differential realization of lie superalgebra b 0 1 new coherent states for the lie superalgebra b 0 1 are presented an inhomogeneous differential realization ihdr of the lie superalgebra is obtained this ihdr may be useful for the study of quasi exactly solvable problems of quantum mechanics

memoryless h sup spl infin controllers for state delayed systems in this note a memoryless h sup spl infin controller for the state delayed system is presented the controller is a delay independent stabilizer for the state delayed system which also reduces the h sup spl infin norm of the closed loop transfer function from the disturbance to the controlled output to a prescribed level the controller can be obtained by solving a modified riccati equation

an approximation formula for the kappa matrix elements of the symplectic algebra sp 6 r a simple approximation formula valid to the high degree of accuracy of the toronto approximation is given for both the off diagonal and diagonal matrix elements of the kappa matrices which transform the bosonic realisations of the symplectic algebra into an orthonormal basis for a unitary irreducible representation of sp 6 r

vector coherent state theory of the noncompact orthosymplectic superalgebras i general theory vector coherent states are defined for the positive discrete series irreducible representations of the non compact orthosymplectic superalgebras osp p 2n r where p 2m or 2m 1 an orthonormal bargmann berezin basis symmetry adapted to osp p 2n r so p sp 2n r so p u n is constructed and used to develop the k matrix theory for osp p 2n r a general method is provided for determining the conditions of existence of star representations and of grade star representations in the osp 2 2n r case and the branching rule for their decomposition into a direct sum of so p sp 2n r irreducible representations as a by product it also enables the matrix elements of the odd generators between basis states of lowest weight so p u n irreducible representations to be calculated in a straightforward way

sample size determination for case control studies and the comparison of stratified and unstratified analyses woolson bean and rojas 1986 biometrics 42 927 932 present a simple approximation of sample size for cochran s 1954 biometrics 10 417 451 test for detecting association between exposure and disease it is useful in the design of case control studies we derive a sample size formula for cochran s statistic with continuity correction which guarantees that the actual type i error rate of the test does not exceed the nominal level the corrected sample size is necessarily larger than the uncorrected one given by woolson et al and the relative difference between the two sample sizes is considerable allocation of equal number of cases and controls within each stratum is asymptotically optimal when the costs per case and control are the same when any effect of stratification is absent cochran s stratified test although valid is less efficient than the unstratified one except for the important case of a balanced design

partially coherent states of the real symplectic group in the present paper we introduce partially coherent states for the positive discrete series irreducible representations d n 2 1 n 2 of sp 2d r encountered in physical applications these states are characterized by both continuous and discrete labels the latter specify the row of the irreducible representation 1 2 d of the maximal compact subgroup u d while the former parametrize an element of the factor space sp 2d r h where h is the sp 2d r subgroup leaving the 1 2 d representation space invariant we consider three classes of partially coherent states respectively generalizing the perelomov and barut girardello coherent states as well as some recently introduced intermediate coherent states we prove that each family of partially coherent states forms an overcomplete set in the representation space of d n 2 1 n 2 and study its generating function properties we show that it leads to a representation of the sp 2d r generators in the form of differential operato

representations of the groups sp n r and sp n in a u n basis the explicit expressions for the infinitesimal operators and the finite matrix elements with respect to a u n basis are obtained for the representations of the most degenerate series of sp n r and for the irreducible unitary representations of sp n with the highest weights m 0 0

induced shift tensors in vector coherent state theory it is shown that the so called shift tensors of biedenharn and louck have a natural expression within the framework of vector coherent state vcs theory vcs theory was developed to give a systematic construction of the representations of lie groups and to derive the corresponding matrices representing the elements of their lie algebras in this paper there are explicit realizations of shift tensors within the framework of vcs theory for semisimple lie groups in terms of shift tensors for suitably defined subgroups

bosonic realisation of sp 4 r and the spectrum of two particle quantum systems a new bosonic holstein primakoff type realisation of sp 4 r is applied in the development of 1 n expansions of non relativistic spin independent two particle hamiltonians as a non trivial example the spectrum of helium like ions is considered and the energies of the low lying levels with given values of total angular momentum are computed the calculation is explicitly performed in an approximation up to the second degree in the small expansion parameter