canonical orthonormal wigner supermultiplet basis the explicit construction of an orthonormal basis for states of good spin isospin and su 4 wigner supermultiplet symmetry is given in a bargmann representation space a complete set of quantum labels is provided by a sp 3 r contains implies u 3 complementary symmetry

sensitivity analysis in plane stress elasto plasticity and elasto viscoplasticity abstract development of computational models for plane stress elasto plasticity is known to make some difficulties non existent in 3d and plane strain formulations tedious derivation of plane stress algorithmic tangent matrices which cannot be directly obtained as a special case of 3d matrices may serve here as an example in this paper specific features of plane stress computations are discussed in the context of parameter sensitivity analysis elasto plastic material model with isotropic kinematic hardening as well as two viscoplastic models are considered explicit expressions for sensitivity gradients are derived instructive examples solved by fem illustrate the paper

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

dual bases and some coupling coefficients for su4 su2 su2 sun son and sp4 u2 general analytical expressions for isoscalar factors reduced wigner coefficients for the couplings p0 0q to 0 and p10 p20 to 0 in the chains su4 su2 su2 and sun son n 3 are derived by using the algebra of complementary groups and methods of analytical continuation a review of alternative approaches to the problem grounded on the concept of biorthogonal systems of non canonical bases is given it is demonstrated that the construction of dual bases polynomial and stretched or related ones is associated with the use of certain bilinear combinations of isoscalar factors and solutions of the boundary value problem for isoscalar factors overlaps are given for different non canonical basis states of the two parametric irreducible representations for the chains su4 su2 su2 sun son and sp4 u2 as well as the expansions of the projected basis states for sp4 u2 in terms of canonical basis states a new expression for some special su2 clebsch gordan coefficients is also given

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

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

shape sensitivity in elasto plastic computations an algorithm for design sensitivity analysis of elasto plastic response for the case of design dependent geometry is discussed the domain parametrization approach is employed some important implications for the concept of isoparametric element formulation are indicated the idea of shape sensitivity computations is similar to that of sensitivity with respect to material parameters however derivation of additional geometric matrices and integral expressions is necessary a possible discontinuity of sensitivity solutions is indicated and it is shown that the algorithm handles it correctly a plane stress computational example illustrates the application of the method

generating kernel for the boson realisation of symplectic algebras the discussion of boson realisations of symplectic algebras requires as an essential ingredient an operator k needed for the passage from the dyson to the holstein primakoff realisation in previous papers the matrix form of k2 was derived through appropriate recursion relations in the present analysis the authors show that k2 can be determined in explicit and closed form as the overlap of coherent states of the symplectic group the matrix elements of k2 can then be obtained by expanding this overlap in terms of appropriate eigenstates

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

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

shape and non shape structural sensitivity analysis for problems with any material and kinematic non linearity abstract the general problem of sizing material selection load parameter and shape sensitivity of non linear structural response is considered the non linear sensitivity path is followed by an incremental strategy employing the continuum based updated lagrangian description the resulting finite element equations are derived both the direct differentiation and adjoint structure strategies are considered the merits of using the consistent tangent matrix for sensitivity calculations are indicated the formulation employs the so called reference volume concept and may be seen as a further attempt to unify the shape and non shape sensitivity analysis procedures

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

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

canonical orthonormal basis for su 3 so 3 ii reduced matrix elements of the su 3 generators for pt i see ibid vol 18 p 1891 1985 a simple algorithm is given for the calculation of reduced matrix elements of the generators of the su 3 algebra in the canonical su 3 contains implies so 3 basis introduced in i

reduction of the unitary group to its orthogonal or symplectic subgroup a unified approach based upon complementary groups a unified analysis of the state labelling problems for the d row irreducible representations of u n when reduced with respect to either its orthogonal subgroup o n or its sympletic subgroup usp n the latter in the even n case is carried out by using appropriate metrics and the complementarity relationship between the group o n and sp 2d r or usp n and so 2d in this way a recently proposed canonical solution to the u n contains implies o n state labelling problem is extended to the u n contains implies usp n chain this shows the equivalence between both these internal state labelling problems and the external state labelling problem for u d as expressed in littlewood s branching rules for u n contains implies o n and u n contains implies usp n

complementarity relation between the usp 2ν and so 2d lie groups the object of this paper consists of finding a complementary group with respect to usp n within either irreducible representation 1 2 dn or 1 2 dn 13 2 of the real symplectic group sp 2dn r for such purpose the author deals with the full orthogonal group o n n 2 nu or 2 nu 1 and the unitary symplectic group usp n n 2 nu in a unified way by introducing an appropriate metric a similar treatment applied to their complementary group determination leads to the well known sp 2d r group in the o n case and to the so 2d group in the usp n case

nonlinear structural design sensitivity analysis for path dependent problems part 2 analytical examples abstract in part 1 of this paper a general theory of design sensitivity analysis for nonlinear structures with history dependent effects is given the theory is also reduced to several special cases to show its generality in the present paper three examples are used to analytically verify the theory both the adjoint structure method asm and the direct differentiation method ddm are demonstrated the examples are 1 reaction force sensitivity for an indeterminate beam 2 displacement sensitivity for several cases of an axial force element and 3 stress and displacement sensitivities for a visco elastic problem these examples provide insights for numerical implementation of the theory for more complex design problems

a steffensen s type method in banach spaces with applications on boundary value problems in this paper a modified steffensen s type iterative scheme for the numerical solution of a system of nonlinear equations is studied two convergence theorems are presented the numerical solution of boundary value problems by the multiple shooting method using the proposed iterative scheme is analyzed

boson realisation of symplectic algebras the boson realisation of the two dimensional symplectic lie algebra sp 2 for a given irreducible representation irrep of the sp 2 group has been known for a long time more recently the boson realisation of the 2d dimensional symplectic lie algebras sp 2d have been derived for particular irreps of the sp 2d group the author outlines the corresponding result for an arbitrary irrep of sp 2d

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

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

geometric interpretations of two branching theorems of d e littlewood abstract d e littlewood proved two branching theorems for decomposing the restriction of an irreducible finite dimensional representation of a unitary group to a symmetric subgroup one is for restriction of a representation of u n to the rotation group so n when the given representation of u n has nonnegative highest weight of depth n 2 it says that the multiplicity in so n of an irreducible representation of so n of highest weight is the sum over of the multiplicities of in the u n tensor product the allowable s being all even nonnegative highest weights for u n littlewood s proof is character theoretic the present paper gives a geometric interpretation of this theorem involving the tensor products explicitly the geometric interpretation has an application to the construction of small infinite dimensional unitary representations of indefinite orthogonal groups and for each of these representations to the determination of its restriction to a maximal compact subgroup the other littlewood branching theorem is for restriction from u 2 r to the rank r quaternion unitary group sp r it concerns nonnegative highest weights for u 2 r of depth r and its statement is of the same general kind the present paper finds an analogous geometric interpretation for this theorem also

coupling coefficients of sun son for on scalar microscopic theory of collective states boson realisations of all on scalar states which span a subspace of the hilbert space for the collective degrees of freedom of the nucleus and are labelled by irreducible representations of the chain of subgroups u3 contains implies u2 contains implies u1 of the complementary group sp 6 r are given two general expressions for isoscalar factors reduced wigner coefficients of sun contains implies son needed for constructing on scalar basis functions depending on the microscopic collective variables in the oscillator basis are derived these isoscalar factors couple three symmetric representations of sun un p10 p20 p30 to the representation h1h2h3 of sun subduced to the son on scalar representation and more generally to the son irreducible representation fff which appears in the case of the closed shells