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

a method for the estimation and identification of transfer function models summary this paper proposes an approach to the simultaneous identification and estimation of transfer function models an estimation procedure is advanced consisting of a sequence of four ordinary least squares calculations and the asymptotic properties of each step are presented consistency and asymptotic efficiency being shown a strategy for the identification of the model structure based on a selection criterion that arises naturally from the last step of the estimation algorithm is also investigated consideration is given to the numerical implementation of the techniques discussed and finally a practical example is employed to illustrate the methods

recursive estimation of autoregressions this paper is concerned with the recursive estimation of autoregressive models in particular the realtime determination of the order of such a specification a model selection criterion based on the minimum description length principle due to rissanen is discussed and strong consistency in the stationary situation is shown alternative criteria are also considered and modifications required to introduce forgetting when the procedures are implemented in the non stationary case are presented some simulation evidence on the performance of the criteria when applied to stationary and non stationary processes is given

fluctuations of extensive functions of quenched random couplings an extensive quantity is a family of functions v of random parameters indexed by the finite regionsv subsets of d over which v are additive up to corrections satisfying the boundary estimate stated below it is shown that unless the randomness is nonessential in the sense that lim v v has a unique value in the absolute i e not just probabilistic sense the variance of such a quantity grows as the volume ofv of particular interest is the free energy of a system with random couplings for such v bounds are derived also for the generating functione et in a separate application variance bounds are used for an inequality concerning the characteristic exponents of directed polymers in a random environment

directed polymers in a random medium 1 d expansion and the n tree approximation the authors develop a method which allows one to obtain 1 d expansions for the problem of directed polymers in a random medium using this method they expand thermal properties such as the free energy the specific heat the overlaps and the fluctuations of the transverse displacement their results are consistent with the existence of a finite upper critical dimension above which the low temperature phase is mean field like i e characterised by a broad distribution of overlaps broken replica symmetry and transverse fluctuations which scale as the length of the polymer

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

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

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

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

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

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

construction of coupling coefficients of su 4 in a supermultiplet basis matrix elements of the unit tensor operators of ranks 1 2 and 1 1 in the non orthonormal draayer basis of su 4 are found the evaluation algorithms for the su 4 coupling coefficients are proposed for coupling one arbitrary and one four six or ten dimensional irreducible representation of su 4 in the su 2 su 2 chain as well as in the case of the semistretched coupling the construction of the overlaps of the draayer basis states is considered

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

determining a portfolio of linear time series models on applique les concepts de degre d evidence suggeres par jeffreys 1961 au probleme de la determination de l ordre dans des modeles de series temporelles lineaires

on the posterior odds of time series models summary this paper is concerned with model selection in time series analysis a bayesian approach is adopted to investigate various decision criteria previously advocated the discussion is conducted in the context of a class of nonlinear time series models of sufficient generality to encompass a wide range of processes currently of interest

fully bayesian analysis of arma time series models abstract statistical analysis of autoregressive moving average arma models is an important non standard problem no classical approach is widely accepted legitimacy for most classical approaches is based solely on asymptotic grounds while small sample sizes are common the only obstacle to the bayesian approach are designing a structure through which prior information can be incorporated and designing a practical computational method the objective of this work is to overcome these two obstacles in addition to the standard results the bayesian approach gives a different method of determining the order of the arma model that is p q

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

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

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

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

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

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

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

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

divergence of shape fluctuations in two dimensions we consider stochastic growth models such as standard first passage percolation on z d where to leading order there is a linearly growing deterministic shape under natural hypotheses we prove that for d 2 the shape fluctuations grow at least logarithmically in all directions although this bound is far from the expected power law behavior with exponent 1 3 it does prove divergence with additional hypotheses we obtain inequalities involving and the related exponent which is expected to equal 2 3 for d 2 combining these inequalities with previously known results we obtain for standard first passage percolation the bounds 1 8 for d 2 and 3 4 for all d

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

sharpness of the phase transition in percolation models the equality of two critical points the percolation threshold ph and the point p where the cluster size distribution ceases to decay exponentially is proven for all translation invariant independent percolation models on homogeneous d dimensional lattices d 1 the analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter m h which for h q reduces to the percolation density p at the bond density p l e in the single parameter case these are 1 m hdm dh m2 mdm d and 2 dm d j mdm dh inequality 1 is intriguing in that its derivation provides yet another hint of a 3 structure in percolation models moreover through the elimination of one of its derivatives 1 yields a pair of ordinary differential inequalities which provide information on the critical exponents and one of these resembles an ising model inequality of fr hlich and sokal and yields the mean field bound 5 2 and the other implies the result of chayes and chayes that an inequality identical to 2 is known for ising models where it provides the basis for newman s universal relation 5 1 1 and for certain extrapolation principles which are now made applicable also to independent percolation these results apply to both finite and long range models with or without orientation and extend to periodic and weakly inhomogeneous systems