reduced notation inner plethysms and the symmetric group the reduced notation for irreducible representations of the symmetric group s is interpreted in terms of symmetric formal series and vertex operators and is used to prove a number of properties of reduced kronecker products and inner plethysms in an n independent manner conditions for self associativity of kronecker products and inner plethysms are established reduced inner plethysms are developed and applied to the question of non simple phase groups among the symmetric sn and alternating an groups

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

linearized oscillation theory for a nonlinear nonautonomous delay differential equation oscillation properties of two following equations are compared a scalar nonlinear delay differential equation y t k 1m rk t f k y hk t 0 with rk t o hk t t and a linear delay differential equation x t k 1m rk t x hk t 0 coefficients rk t and delays are not assumed to be continuous as an application explicit oscillation and nonoscillation conditions are established for nonlinear equations arising in population dynamics

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

linearized oscillation of first order nonlinear neutral delay differential equations abstract in this paper we establish a equivalence of oscillation for the first order nonlinear neutral delay differential equation x t px t i 1 m q i f i x t i 0 and its corresponding linear equation x t px t i 1 m q i x t i 0 under a weaker condition on fi

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

linearized oscillation theory for a nonlinear delay impulsive equation for a scalar nonlinear impulsive delay differential equation y t k 1m rk t fk y hk t 0 t j y j ij y j with rk t 0 hk t t limj j such an auxiliary linear impulsive delay differential equation x t k 1m rk t ak t x hk t 0 x j bjx j is constructed that oscillation nonoscillation of the nonlinear equation can be deduced from the corresponding properties of the linear equation coefficients rk t and delays are not assumed to be continuous explicit oscillation and nonoscillation conditions are established for some nonlinear impulsive models of population dynamics such as the impulsive logistic equation and the impulsive generalized lasota wazewska equation which describes the survival of red blood cells it is noted that unlike nonimpulsive delay logistic equations a solution of a delay impulsive logistic equation may become negative

identifying gait asymmetry using gyroscopes a cross correlation and normalized symmetry index approach abstract injury to a lower limb may disrupt natural walking and cause asymmetrical gait therefore assessing the gait asymmetry has become one of the important procedures in gait analysis this paper proposes the use of wireless gyroscopes as a new instrument to determine gait asymmetry it also introduces two novel approaches normalized cross correlations cc norm and normalized symmetry index si norm cc norm evaluates the waveform patterns generated by the lower limb in each gait cycle si norm provides indications on the timing and magnitude of the bilateral differences between the limbs while addressing the drawbacks of the conventional methods one way anova test reveals that cc norm can be considered as single value indicator that determines the gait asymmetry p si norm in asymmetrical gait were different from normal gait si norm in asymmetrical gait were found to be approximately 20 greater than si norm in normal gait during pre swing and initial swing

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

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

the symmetric group branching rules products and plethysms for spin representations it is shown that the symmetric group sn may be usefully embedded in the orthogonal group on and that this embedding leads directly to an n independent reduced notation for both the spin and ordinary representations of sn making use of this embedding together with the properties of q functions or hall littlewood functions branching rules for on spin down sn are developed and the general rule for the decomposition of spin representations under sn spin down sn 1 is obtained simple methods are given for calculating all possible kronecker products involving the spin and ordinary representations of sn and the resolution of kronecker squares into their symmetric and antisymmetric parts the spin representations of sn are systematically classified as to their orthogonal symplectic or complex characters the emphasis throughout is on obtaining results that obviate the need for explicit character tables and presenting results in an n independent manner as much as possible

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

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

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

oscillation theorems for first order nonlinear neutral functional differential equations abstract this paper discusses a class of first order nonlinear neutral differential equations with variable coefficients and variable deviations sharp conditions are established for all solutions of the equations to be oscillatory linearized oscillation criteria of the equations are also given

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

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

linearized oscillations for odd order neutral delay differential equations abstract consider the n th order nonlinear neutral delay differential equation d n dt n x t p t g x t q t h x t 0 1 where n 1 is an odd integer we prove that under appropriate hypotheses eq 1 oscillates provided that the same is true for an associated linear equation with constant coefficients of the form d n dt n y t p 0 y t q 0 y t 0 a partial converse is also presented where we show that under appropriate hypotheses eq 1 has a positive solution

ambulatory measurement of 3d knee joint angle three dimensional measurement of joint motion is a promising tool for clinical evaluation and therapeutic treatment comparisons although many devices exist for joints kinematics assessment there is a need for a system that could be used in routine practice such a system should be accurate ambulatory and easy to use the combination of gyroscopes and accelerometers i e inertial measurement unit has proven to be suitable for unrestrained measurement of orientation during a short period of time i e few minutes however due to their inability to detect horizontal reference inertial based systems generally fail to measure differential orientation a prerequisite for computing the three dimentional knee joint angle recommended by the internal society of biomechanics isb a simple method based on a leg movement is proposed here to align two inertial measurement units fixed on the thigh and shank segments based on the combination of the former alignment and a fusion algorithm the three dimensional knee joint angle is measured and compared with a magnetic motion capture system during walking the proposed system is suitable to measure the absolute knee flexion extension and abduction adduction angles with mean sd offset errors of 1 degree 1 degree and 0 degrees 0 6 degrees and mean sd root mean square rms errors of 1 5 degrees 0 4 degrees and 1 7 degrees 0 5 degrees the system is also suitable for the relative measurement of knee internal external rotation mean sd offset error of 3 4 degrees 2 7 degrees with a mean sd rms error of 1 6 degrees 0 5 degrees the method described in this paper can be easily adapted in order to measure other joint angular displacements such as elbow or ankle

linearized oscillation of odd order nonlinear neutral delay differential equations i abstract in this paper we proved that the odd order nonlinear neutral delay differential equation x t p t g x t n q t h x t 0 has the same oscillatory character as its linearized equation x t p 0 x t n q 0 x t 0 under some rather relaxed conditions on g u and h u where p 0 lim t p t q 0 lim t q t

symmetrised kronecker products of the fundamental representation of sp n r the role played by s function series in the character theory of the non compact symplectic group sp n r is exploited to derive general formulae for the evaluation of symmetrised kronecker products of the fundamental sp n r representation applications of the results to sp 3 r and their impact on the nuclear symplectic shell model are discussed for simple cases

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

some noteworthy spin plethysms the spin plethysms lambda g x delta that arise in the reduction of delta under so n to g when 1 to lambda g are considered it is shown that for the simple lie algebras of rank k if lambda g phi g the adjoint representation of g then phi g x delta 2 k2 delta g where delta g is the representation of g whose highest weight is half the sum of the positive roots certain results for other representations are described a remarkable series of s functions is introduced leading to a new dimensional equality between certain representations of o 2k and sp 2k

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

necessary and sufficient condition for oscillation of a neutral differential system with several delays abstract in this paper we prove that a general neutral differential system with several delays has a nonoscillatory solution if and only if its characteristic equations has a real root

linearized oscillations for even order neutral differential equations abstract we are concerned with the oscillation of all bounded solutions of some even order linear neutral delay differential equations we establish a comparison theorem and a linearized oscillation result and we prove the existence of a bounded positive solution

plethysms of schur functions and the shell model we present a method for evaluating plethysms of schur functions that is conceptually simpler than existing methods moreover the algorithm can be easily implemented with an algebraic computer language plethysms of sums differences and products of s functions are dealt with in exactly the same manner as plethysms of simple s functions sums and differences of s functions are of importance for the description of multi shell configurations in the shell model the number of variables in which the s functions are expressed can be specified in advance significantly simplifying the calculations in typical applications to many body problems the method relies on an algorithm that we have developed for the product of monomial symmetric functions we present a new way of calculating the kostka numbers using gel fand patterns and give as well a new formula for the littlewood richardson coefficients

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

a dimension formula for reduced plethysms the boson calculus formalism is used to construct realizations of basis states of irreducible representations of unitary groups taking as a paradigm the interacting boson models of atomic nuclei these realizations together with a theorem on plethysms for obtaining branching rules allowed us to obtain a dimension formula for reduced plethysms

accuracy and precision of hind limb foot contact timings of horses determined using a pelvis mounted inertial measurement unit abstract gait analysis using small sensor units is becoming increasingly popular in the clinical context in order to segment continuous movement from a defined point of the stride cycle knowledge about footfall timings is essential we evaluated the accuracy and precision of foot contact timings of a defined limb determined using an inertial sensor mounted on the pelvis of ten horses during walk and trot at different speeds and in different directions foot contact was estimated from vertical velocity events occurring before maximum sensor roll towards the contralateral limb foot contact timings matched data from a synchronised hoof mounted accelerometer well when velocity minimum was used for walk mean sd difference of 15 18 ms across horses and velocity zero crossing for trot mean sd difference from 4 14 to 12 7 ms depending on the condition the stride segmentation method also remained robust when applied to movement data of hind limb lame horses in future this method may find application in segmenting overground sensor data of various species

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

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