multifractality of nonlinear iterative processes the havlin bunde multifractal hypothesis physica d 38 184 1989 is expanded in the form of the dimension invariance approach to nonlinear iterative recursion processes such as dielectric breakdown phase transitions from periodic attractors to chaos and cascades in turbulence comparison with model and laboratory data of different authors shows that for strong nonlinearity the dimension invariance is broken

vector coherent state theory of the noncompact orthosymplectic superalgebras ii some selected examples for pt i see ibid vol 23 p 5383 1990 the techniques presented in the first paper of the present series for determining the conditions for the existence of star or grade star positive discrete series irreducible representations of osp p 2n r p 2m or 2m 1 and the branching rule for their decomposition into a direct sum of so p sp 2n r irreducible representations as well as for constructing explicit matrix realizations are illustrated with a few selected examples the latter include the most general irreps of osp 1 2n 2 osp 2 2 r osp 3 2 r osp 4 2 r osp 2 4 r and the most degenerate irreps of osp 2 2n r in addition all the information necessary for dealing with other cases amenable to a full analytic treatment is provided

embedded surfaces of arbitrary genus minimizing the willmore energy under isoperimetric constraint the isoperimetric ratio of an embedded surface in mathbb r 3 is defined as the ratio of the area of the surface to power three to the squared enclosed volume the aim of the present work is to study the minimization of the willmore energy under fixed isoperimetric ratio when the underlying abstract surface has fixed genus g geqq 0 the corresponding problem in the case of spherical surfaces that is g 0 was recently solved by schygulla see schygulla arch ration mech anal 203 901 941 2012 with different methods

analogies between finite dimensional irreducible representations of so 2n and infinite dimensional irreducible representations of sp 2n r i characters and products the analogy between the finite dimensional spin representation of so 2n and the infinite dimensional representation of sp 2n r is made precise it is then shown that this analogy can be extended so as to provide a precise link between each finite dimensional unitary irreducible representation of so 2n and a corresponding infinite dimensional unitary irreducible representation of sp 2n r the analogy shows itself at the level of the corresponding characters and difference characters and involves the use of schur function methods to express both characters and difference characters of so 2n and sp 2n r in terms of characters of irreducible representations of their common subgroup u n the analogy is extended still further to cover the explicit decomposition of not only tensor products of and with other unitary irreducible representations of so 2n and sp 2n r respectively but also arbitrary tensor powers of and

schur function analysis of the unitary discrete series representations of the non compact symplectic group the authors develop and analyse certain character reductions for the infinite dimensional unitary discrete series representations of the non compact symplectic group sp 2n r the group reductions considered are sp 2k r contains implies sp 2 r o k and the more general sp 2nk r contains implies sp 2n r o k they use schur function techniques to derive succinct formulae involving certain infinite series of schur functions the results are relevant to the study of many body systems with interactions of bilinear form and to the description of various quantum phenomena including collective behaviour

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

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

analogies between finite dimensional irreducible representations of so 2n and infinite dimensional irreducible representations of sp 2n r ii plethysms the basic spin difference character of so 2n is a useful device in dealing with characters of irreducible spinor representations of so 2n it is shown here that its kth fold symmetrized powers or plethysms associated with partitions of k factorize in such a way that r where r is the frobenius rank of the analogy between so 2n and sp 2n r is shown to be such that the plethysms of the basic harmonic or metaplectic character of sp 2n r factorize in the same way to give r moreover the analogy is shown to extend to the explicit decompositions into characters of irreducible representations of so 2n and sp 2n r not only for the plethysms themselves but also for their factors and explicit formulas are derived for each of these decompositions expressed in terms of various group subgroup branching rule multiplicities particularly those defined by the restriction from o k to the symmetric group sk illustrative examples are included as well as a

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

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

s function series revisited a systematic and readily programmable computational method is given to extract the s function content of any arbitrary symmetric polynomial function applications to the derivation of new series and the outer product or skew division of an s function by a series are shown

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

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

complementarity relation between the u p q and u n lie groups and some applications to atomic physics the complementarity relation between the unitary groups u d and u n within the symmetrical irreducible representations of the larger unitary group u dn is extended to non compact groups it is proved that the pseudo unitary group u p q is complementary with respect to u n within some positive discrete series irreducible representations of the larger pseudo unitary group u pn qn the latter arise when reducing the metaplectic irreducible representations 1 2 dn and 1 2 dn 13 2 of the real symplectic group sp 2dn r where d p q and they are characterised by a single label the eigenvalue of the first order casimir operator some applications of the u p q u n complementarity to atomic physics are outlined for such purposes the isomorphism between the lie algebras of su 2 2 and so 4 2 is used extensively the mathematical framework underlying the kibler negadi approach 1983 4 of the hydrogen atom dynamical group is extended to the independent electron dynamical group of intrashell many electron states as well as to the correlated electron dynamical group of intrashell doubly excited states

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

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

spectrum generating functions for strings and superstrings the spectrum generating functions for a variety of strings and superstrings are derived these are expressed in terms of characters of irreducible representations of the appropriate transverse spacetime symmetry group and in the case of heterotic strings the relevant gauge group e8 e8 or spin 32 z2 the expansions involve principal specialisations of characters of both gl n and gl m n in the limit as m and n tend to infinity by using schur function methods modification rules and various theta function identities a number of spectrum generating functions are derived and tabulated the equivalence of the gso projected neveu schwarz ramond superstring and the green schwarz superstring is proved and the connection between the lattice formulation and the gso projected neveu schwarz ramond formulation of heterotic strings is spelled out

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

matrix elements of operators in symmetrical u 6 u 3 u 2 u 1 and u 6 su 3 so 3 so 2 basis in this paper we consider a basis for n boson states classified by the chain of groups u 6 u 3 u 2 u 1 we determine analytical expressions for the matrix elements of the boson creation and annihilation operators as well as for the u 6 generators with respect to that basis for such purpose we use the wigner eckart theorem with respect to u 3 and calculate the reduced matrix elements of the appropriate irreducible tensors from their matrix elements between highest weight states then we apply the transformation brackets from the chain u 6 u 3 u 2 u 1 to the physical chain u 6 su 3 so 3 so 2 to obtain the matrix elements of the above mentioned operators in a basis corresponding to the latter the matrix elements so determined can be used either in the nuclear interacting boson model of arima and iachello or in the microscopic nuclear collective model of vanagas et al

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

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

three dimensional turbulent diffusion versus critical phenomena the critical exponent nu of the correlated or coherent length of a turbulent cluster has been related to the minimal fractal dimension dmin of multifractal isotropic turbulence the relationship has turned out to be nu dmin 3 2 for the homogeneous case dmin 3 and hence nu 1 2 the so called mean field approach of the theory of critical phenomena for dmin approximately 2 36 the well known turbulent value nu approximately 0 63 this result allows one to classify this case as the critical phenomenon of the so called thermal class of universality transition to the percolation class of universality nu approximately 0 9 is determined by the boundary conditions

the representation space of the nuclear symplectic sp 6 r shell model the problem of obtaining the sp 6 r irreducible representations that arise from the symmetrized products of a copies of the fundamental representation of sp 6 r is considered a simple method is developed that rapidly leads to the determination of the sp 6 r content up to a prescribed cut off in practical calculations a is identified with the nucleon number it is shown that for a given excitation energy cut off n omega and sufficiently large a the number and multiplicities of the sp 6 r irreducible representations is fixed and the content can be expressed in an a independent manner complete results are given for n or 16 there is no difficulty in principle in going to higher excitations and details are given for determining the sp 6 r content for a or 16

products and symmetrized powers of irreducible representations of the calculation of branching rules tensor products and plethysms of the infinite dimensional harmonic series unitary irreducible representations of the non compact group is considered and the duality between and sp 2k exploited the branching rule for the restriction of an arbitrary harmonic series irreducible representation of to u n is derived and the decomposition is given explicitly for each of the infinite number of fundamental harmonic series irreducible representations of whose direct sum constitutes the metaplectic representation h of a concise expression for the decomposition of tensor products is derived and a complete analysis of the terms in both and is given a general formula for plethysms of arbitrary irreducible representations of is derived and its implementation illustrated both by means of a detailed generic example and by a complete determination of the symmetric and antisymmetric terms of finally relationships that arise from the embedding of the product groups and in the metaplectic group mp 4nk are discussed

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 new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

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

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

determination of the sp 2d r generator matrix elements through a boson mapping for the discrete series irreducible representations lambda n 2 d of sp 2d r the determination of the sp 2d r generator matrix elements in an sp 2d r contains implies u d basis is reduced to the much simpler calculation of boson operator matrix elements between u nu contains implies u d boson states where nu d d 1 2 the key of this reduction is the previously derived holstein primakoff boson representation of the sp 2d r generators as an illustration the case of sp 6 r is worked out in detail

raising operators for the osp 1 2n r orthosymplectic lie superalgebras the branching rule is obtained for the decomposition of the osp 1 2n r oscillator like unitary irreducible representations into sp 2n r representations raising operators for osp 1 2n r sp 2n r are determined and used to construct a basis in the irreducible representation carrier space the matrix elements of the odd generators between two sp 2n r lowest weight states are calculated the general results so obtained are illustrated with the osp 1 4 r example

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

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

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

plethysm for the non compact group and new s function identities methods of computing plethysms of the fundamental unitary irreducible representations of the non compact symplectic group sp 2n r are considered complete results are given for the symmetrized second powers a number of new s function identities are reported the stability properties of the sp 2n r plethysms are noted as well as a remarkable conjugacy relation the application of the plethysms to n particles in an isotropic harmonic oscillator is briefly outlined

multifractal percolation and growth in intermittent media the havlin bunde multifractal hypothesis for the probability density of a random walker is used to obtain the scaling law of thepth order correlation function of the concentration for percolation and of the height for growing surfaces differences c p r r p in intermittent media it is shown that near the transition to homogeneity p a p in p po wherea andp0 are some constants good agreement with recent experiments and computer simulations of different authors is established

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

a class of unitary representations of the lie group sp 3 r its coherent states and its map to a symplectic realization on sp 3 r unitary representations from the positive discrete series of sp 3 r with lowest weight w w0w0w0 are considered by use of new relations on the enveloping algebra the generators are constructed as differential operators acting on functions of six real variables coherent states for these representations are constructed with the help of the iwasawa decomposition and used to map the representation space to a symplectic realization on the dual sp 3 r

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

min max theory and the willmore conjecture in 1965 t j willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 we prove this conjecture using the min max theory of minimal surfaces

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

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

new s function series and non compact lie groups a number of new infinite series of s functions are described in terms of their generating functions and s function content applications to the character theory of non compact lie groups are noted

slinkies and the s function content of certain generating functions a variety of methods are described for expanding particular generating functions in terms of their s function content plethysms are used to establish conjugacy relations between s function series and the task of evaluating certain products of classical s function series introduced by littlewood 1950 is then completed slinky diagrams are introduced to represent non standard s functions and their modification to give standard s functions the role of slinkies in determining the s function content of other generating functions is then explained and exemplified

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

a gap theorem for willmore tori and an application to the willmore flow abstract in 1965 willmore conjectured that the integral of the square of the mean curvature of a torus immersed in r 3 is at least 2 2 and attains this minimal value if and only if the torus is a mobius transform of the clifford torus this was recently proved by marques and neves 2012 in this paper we show for tori that there is a gap to the next critical point of the willmore energy and we discuss an application to the willmore flow we also prove an energy gap from the clifford torus to surfaces of higher genus

products and symmetrized powers of irreducible representations of and their associates the calculation of kronecker products and plethysms of the infinite dimensional harmonic series unitary irreducible representations of the non compact group is considered the complementarity of and o k is used to define associate irreducible representations of this leads to simple relationships between kronecker products and plethysms of irreducible representations of and those of their corresponding associate irreducible representations in the process of proving the validity of these previously conjectured relationships several new identities are found for plethysms involving infinite series of schur functions in addition a general formula for plethysms of arbitrary irreducible representations of is derived and its implementation is illustrated with a detailed example a remarkable analogy is then observed between plethysms of the basic harmonic irreducible representations of and those of the basic spin irreducible representations of so 2n

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

stable branching rules for classical symmetric pairs we approach the problem of obtaining branching rules from the point of view of dual reductive pairs specifically we obtain a stable branching rule for each of 10 classical families of symmetric pairs in each case the branching multiplicities are expressed in terms of littlewood richardson coefficients some of the formulas are classical and include for example littlewood s restriction rule as a special case

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

the paraboson fock space and unitary irreducible representations of the lie superalgebra mathfrak osp 1 2n it is known that the defining relations of the orthosymplectic lie superalgebra mathfrak osp 1 2n are equivalent to the defining triple relations of n pairs of paraboson operators b pm i in particular with the usual star conditions this implies that the parabosons of order p correspond to a unitary irreducible infinite dimensional lowest weight representation v p of mathfrak osp 1 2n apart from the simple cases p 1 or n 1 these representations had never been constructed due to computational difficulties despite their importance in the present paper we give an explicit and elegant construction of these representations v p and we present explicit actions or matrix elements of the mathfrak osp 1 2n generators the orthogonal basis vectors of v p are written in terms of gelfand zetlin patterns where the subalgebra mathfrak u n of mathfrak osp 1 2n plays a crucial role our results also lead to character formulas for these infinite dimensional mathfrak osp 1 2n representations furthermore by considering the branching mathfrak osp 1 2n supset mathfrak sp 2n supset mathfrak u n we find explicit infinite dimensional unitary irreducible lowest weight representations of mathfrak sp 2n and their characters

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

the large genus limit of the infimum of the willmore energy the willmore energy of a surface immersed into bbb r n is the integral of its squared mean curvature it is known that the infimum beta n p of the willmore energy among all closed oriented surfaces of genus p in bbb r n is attained by a smooth embedded surface and that 4 pi beta n p 8 pi for p geq 1 we show that beta n p converges to 8 pi as p goes to infinity

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

new branching rules induced by plethysm we derive group branching laws for formal characters of subgroups of leaving invariant an arbitrary tensor t of young symmetry type where is an integer partition the branchings and fixing a vector vi a symmetric tensor gij gji and an antisymmetric tensor fij fji respectively are obtained as special cases all new branchings are governed by schur function series obtained from plethysms of the schur function s by the basic m series of complete symmetric functions and the l m 1 series of elementary symmetric functions our main technical tool is that of hopf algebras and our main result is the derivation of a coproduct for any schur function series obtained by plethysm from another such series therefrom one easily obtains generalized newell littlewood formulae and the algebra of the formal group characters of these subgroups is established concrete examples and extensive tabulations are displayed for and showing their involved and nontrivial representation theory the nature of the subgroups is shown to be in general affine and in some instances non reductive we discuss the complexity of the coproduct formula and give a graphical notation to cope with it we also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly nontrivial 2 cocycles the algebra of subgroup characters is identified as a cliffordization of the algebra of symmetric functions for formal characters modification rules are beyond the scope of the present paper but are briefly discussed

simple unified proofs of four duality theorems duality relationships between the irreps irreducible representations of pairs of distinct commuting groups g1 and g2 on hilbert spaces of interest have long played important roles in the atomic and nuclear shell models in addition to the well known schur weyl duality the most widely used duality relationships are the so called unitary unitary orthogonal symplectic i e noncompact symplectic symplectic symplectic compact symplectics and orthogonal orthogonal dualities proofs of these dualities exist in the literature but most of them are not readily accessible to physicists or give little insight into how they might be used in practice this paper presents unified proofs of the above mentioned dualities based on the explicit construction of states which are simultaneously of extreme weight for the actions of both g1 and g2 the proofs expressed in language familiar to physicists are simple systematic and provide useful insights

hilbert spaces of analytic functions and representations of the positive discrete series of sp 6 r hilbert spaces of analytic functions are constructed which carry irreducible representations of the positive discrete series 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

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

products and plethysms for the fundamental harmonic series representations of u p q we give the decomposition of the kronecker products and the symmetrized kronecker squares of all the fundamental representations of the harmonic series of unitary irreducible representations of u p q the results for u 2 2 are relevant to two electron hydrogenic like atoms

the hopf algebra structure of the character rings of classical groups the character ring of covariant irreducible tensor representations of the general linear group admits a hopf algebra structure isomorphic to the hopf algebra of symmetric functions here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions we show that and also admit natural hopf algebra structures that are isomorphic to that of and hence to the isomorphisms are determined explicitly along with the specification of standard bases for and analogous to those used for a major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic schur hall scalar products significantly the adjoint with respect to multiplication no longer coincides as it does in the case with a foulkes derivative or skew operation the adjoint and foulkes derivative now require separate definitions and their properties are explored here in the orthogonal and symplectic cases moreover the hopf algebras and are not self dual the dual hopf algebras and are identified finally the hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified

symmetrized squares and cubes of the fundamental unirreps of we establish the complete decompositions of the symmetrized kronecker squares of the basic harmonic representations of conjectured by grudzinski and wybourne grudzinski k and wybourne b g 1996 j phys a math gen 29 6631 41 and give analogous complete results for the symmetrized cubes equivalence relationships are shown to exist between certain pairs of plethysms

analytical expressions for the matrix elements of the non compact symplectic algebra analytic matrix elements are derived for all the lowest weight representations of the noncompact symplectic algebra sp 3 r it is shown that the expressions are exact for representations of the type sigma 1 sigma 2 sigma 3 and for all states of an arbitrary representation that are multiplicity free with respect to the u 3 subalgebra furthermore they are remarkably accurate in general

probability densities of random walks in random systems abstract we review recent results for the probability distribution of random walkers in random systems where diffusion is anomalous and the mean square displacement scales with time as r 2 t t 2 w d w 2 the random systems are characterized by structural disorder and by random transition rates in general the mean distribution function p r t of the random walkers is a stretched gaussian and scales as log p r t p r 0 r r t u where u d w d w 1 on random fractals the fluctuations of the density distribution p r t for fixed distance r and time t have a broad logarithmic distribution the average moments p q scale in a multifractal way as p q where q q l space the fluctuations of p are narrow and p q p q