cauchy transform and poisson s equation let u 2 w 2 p 0 1 6 p 6 1 be a solution of the poisson equationu h h 2 l p in the unit disk it is proved that kr uklp 6 apkhklp with sharp constant ap for p 1 and p 1 and that k uklp 6 bpkhklp with sharp constant bp for p 1 p 2 and p 1 in addition is proved that for p 2 jj ujj l1 6 cpkhklp and jjrujj l1 6 cpkhklp with sharp constants cp and cp an extension to smooth jordan domains is given these problems are equivalent to determining the precise value of l p norm of cauchy transform of dirichlet s problem

the probability distribution of the partition function of the random energy model the authors give the expression for both integer and non integer moments of the partition function z of the random energy model in the thermodynamic limit they find that the probability distribution p z can be decomposed into two parts for log z log z finite the distribution is independent of n the size of the system whereas for log z log z positive and of order n the distribution is gaussian these two parts match in the region 1 log z log z n where the distribution is exponential

inverse branching rules a method is given for inverting branching rules which is based on a technique for computing kronecker products due to king 1981 this approach simplifies the calculation of plethysms and branching rules of representations of exceptional groups it may also be applied to basic spin plethysms

synchronization via pinning control on general complex networks this paper studies synchronization via pinning control on general complex dynamical networks such as strongly connected networks networks with a directed spanning tree weakly connected networks and directed forests a criterion for ensuring network synchronization on strongly connected networks is given it is found that the vertices with very small in degrees should be pinned first in addition it is shown that the original condition with controllers can be reformulated such that it does not depend on the form of the chosen controllers which implies that the vertices with very large out degrees may be pinned then a criterion for achieving synchronization on networks with a directed spanning tree which can be composed of many strongly connected components is derived it is found that the strongly connected components with very few connections from other components should be controlled and the components with many connections from other components can achieve synchronization even without controls

pinning control of complex dynamical networks with general topology abstract recently the researches on pinning control of complex dynamical networks have mainly focused on such networks with very specific coupling schemes e g symmetric coupling uniform coupling and linear coupling however most real networks often consist of local units which interact with each other via asymmetric and heterogeneous connections in this paper pinning control of a continuous time complex dynamical network with general coupling topologies is studied some generic stability criteria based on master stability function msf are derived for such a general controlled network which guarantee that the whole network can be pinned to its equilibrium by placing feedback control only on a small fraction of nodes then these results are extended to discrete time case previous results about symmetric uniform or linear coupled networks in this area are included as special cases of the present work numerical simulations of directed networks with weighted coupling pinned by specifically selective pinning scheme are given for illustration and verification

position space renormalisation group for directed diffusion limited aggregation the author uses a position space renormalisation group transformation to show that directed diffusion limited aggregation and directed animals are in different universality classes

on hormander s solution of the bar partial equation i we explain how hormander s classical solution of the bar partial equation in the plane with a weight which permits growth near infinity carries over to the rather opposite situation when we ask for decay near infinity here however a natural condition on the datum needs to be imposed the condition is not only natural but also necessary to have the result at least in the fock weight case the norm identity which leads to the estimate is related to general area type results in the theory of conformal mappings

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

weighted bergman spaces and the integral means spectrum of conformal mappings the classical theory of conformal mappings involves best possible pointwise estimates of the derivative thus supplying a measure of the extremal expansion contraction possible for a conformal mapping it is natural to consider also the integral means of phi t along circles z r where phi is the conformal mapping in question and t is a real parameter 0 1 of the integral means which follows from the classical pointwise estimates is by far too fast better estimates were found by clunie makarov pommerenke bertilsson shimorin and others here we introduce a new method based on area type estimates which discards as little as possible of the information supplied by the area methods the result is a considerable improvement in the estimates of the integral means spectrum known tip to this point

pinning synchronization of fractional order delayed complex networks with non delayed and delayed couplings abstractthis paper considers the global synchronization of fractional order delayed complex networks with non delayed and delayed couplings based on pinning control with the help of comparison theorems for fractional order linear systems with multiple time delays some novel and useful synchronisation criteria are presented by adding linear error feedback controllers to certain selected nodes here the coupling matrices need not be symmetric or irreducible what is more it is shown that the model and results in the paper are more general than those in the literature a numerical example is given to demonstrate the effectiveness and feasibility of the obtained results

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

a generalised self avoiding walk the authors study a generalised self avoiding walk on a lattice in which each vertex may be visited less than k time turban 1983 has argued that for this model the critical exponents should change continuously with k from the standard self avoiding walk exponents k 2 to mean field exponents by generating series expansions for both two and three dimensional lattices the authors find strong support for the conventional and opposing view that the critical exponents remain unchanged for finite k only taking on their mean field values in the limit k to infinity

pinning synchronization of delayed complex dynamical networks with nonlinear coupling abstract in this paper we find that complex networks with the watts strogatz or scale free ba random topological architecture can be synchronized more easily by pin controlling fewer nodes than regular systems theoretical analysis is included by means of lyapunov functions and linear matrix inequalities lmi to make all nodes reach complete synchronization numerical examples are also provided to illustrate the importance of our theoretical analysis which implies that there exists a gap between the theoretical prediction and numerical results about the minimum number of pinning controlled nodes

fluctuations in the random link matching problem using the replica approach and the cavity method we study the fluctuations of the optimal cost in the random link matching problem by means of replica arguments we derive the exact expression of its variance moreover we study the large deviation function deriving its expression in two different ways namely using both the replica method and the cavity method

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

extended poincare supersymmetry rotation groups and branching rules the decomposition of the basic spin irreps of the special orthogonal group so2k into irreps of sod 2 k where d is the spacetime dimension of the extended d dimensional poincare supersymmetry and k is the appropriate automorphism group is determined general results for d 10 several new branching rules for subgroups of so2k are developed a number of specific results are tabulated

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

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

boundary properties of green functions in the plane we study the boundary properties of the green function of bounded simply connected domains in the plane essentially this amounts to studying the conformal mapping taking the unit disk onto the domain in question our technique is inspired by a 1995 article of jones and makarov 11 the main tools are an integral identity as well as a uniform sobolev embedding theorem the latter is in a sense dual to the exponential integrability of marcinkiewicz zygmund integrals we also develop a grunsky identity which contains the information of the classical grunsky inequality this grunsky identity is the case where p 2 of a more general grunsky identily for l p spaces

projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control abstract this paper investigates the projective cluster synchronization of fractional order coupled delay complex network via adaptive pinning control the pinning control method is employed in designing controllers for guaranteeing projective cluster synchronization based on the lyapunov stability theory and well known barbalat s lemma some new sufficient conditions are derived to guarantee the projective cluster synchronization of the fractional order complex network a numerical simulation is given to verify the effectiveness of the theoretical results

upper bounds for multiplicities in a tensor product of representations and in a restriction of group representations to a subgroup upper bounds for multiplicities of irreducible representations of a semisimple compact lie group in the tensor product of representations and multiplicities of irreducible representations of a subgroup in an irreducible representation of a group are derived pairs of groups and subgroups appear from the representation theory of non compact groups

vertex operators and composite supersymmetric s functions the s function basis for the fock space of vertex operator constructions is considered where states are associated with partitions mod lambda the authors show that the matrix element lambda mod valpha z mod mu is essentially given by the composite s function slambda mu z where the argument z identical to z 1 is replicated mod alpha mod times matrix elements of products of vertex operators lambda mod valpha z v beta w mod mu are given up to functions of z w independent of lambda and mu by the composite supersymmetric s function slambda mu z w where the z w are replicated mod alpha mod mod beta mod times the supersymmetric arguments being those associated with negative roots

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

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

projective cluster synchronization in drive response dynamical networks abstract in this paper we study the projective cluster synchronization in a drive response dynamical network with 1 n coupled partially linear chaotic systems because the scaling factors characterizing the dynamics of projective synchronization remain unpredictable pinning control ideas are adopted to direct the different scaling factors onto the desired values it is also shown that the projection cluster synchronization can be realized by controlling only one node in each cluster numerical simulations on the chaotic lorenz system are illustrated to verify the theoretical results

second order leader following consensus of nonlinear multi agent systems via pinning control abstract without assuming that the interaction diagraph is strongly connected or contains a directed spanning tree this paper studies the second order leader following consensus problem of nonlinear multi agent systems with general network topologies based on graph theory matrix theory and lasalle s invariance principle a pinning control algorithm is proposed to achieve leader following consensus in a network of agents with nonlinear second order dynamics furthermore a pinning consensus protocol is developed for coupled double integrators with a constant reference velocity in particular this paper addresses what kind of agents and how many agents should be pinned and establishes some sufficient conditions to guarantee that all agents asymptotically follow the virtual leader numerical simulations are given to verify the theoretical analysis

on phase diagrams for directed percolation problems the authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which differing from previous phenomenological or small cell real space renormalisation groups is sensitive to the existence of an infinite cluster along any lattice direction this enables them to explore new aspects of the phase diagram for the problem which were not accessible through existing treatments in particular the crossover exponent between one and two dimensional behaviour is shown to be exactly one for directed percolation numerical estimates are in agreement with known inequalities

fractional market dynamics a new extension of a fractality concept in nancial mathematics has been developed we have introduced a new fractional langevin type stochastic dierential equation that diers from the standard langevin equation i by replacing the rst order derivative with respect to time by the fractional derivative of order and ii by replacing white noise gaussian stochastic force by the generalized shot noise each pulse of which has a random amplitude with the stable levy distribution as an application of the developed fractional non gaussian dynamical approach the expression for the probability distribution function pdf of the returns has been established it is shown that the obtained fractional pdf ts well the central part and the tails of the empirical distribution of s p 500 returns c 2000 elsevier science b v all rights reserved

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 theory of angular momenta and higher su n symmetries the authors discuss interrelations between generating invariants gi for the reduction pi mi 1 d pi to d on 1 of the m fold tensor product of su n irreducible representations irreps d pi and polynominal bases of the su m irreps d on 1j om n 1 ok 0 0 0 k times a realisation of the su m irrep d oj om 3 bases is given in terms of gi for su 2 coupling wigner coefficients as a byproduct an expression is obtained for su 2 6 j symbols in terms of only two wigner coefficients the authors also discuss some possibilities of the analysis involved in solving the wigner biedenharn problem construction of orthonormal sets of the wigner coefficients for su n groups n or 3

some recent results on symmetries of lagrangian systems re examined hojman s results 1984 on alternative lagrangians and first integrals related to symmetries of lagrangian systems are shown to be covered by previous treatments in a more geometrical setting it is argued that the use of lagrangians linear in the accelerations for second order systems does not widen the spectrum of results

cluster synchronization in directed networks of non identical systems with noises via random pinning control abstract this paper is concerned with the issue of mean square cluster synchronization in directed networks which consist of non identical nodes infected by communication noises the pinning control method is employed in designing controllers for guaranteeing cluster synchronization meanwhile all the controllers are supposed to occur with different probabilities by introducing the bernoulli stochastic variables based on the lyapunov stability theory and the stochastic theory the sufficient synchronization conditions are derived and proved theoretically which are mainly for the controllers to be designed and the noise intensities finally some numerical examples are presented to demonstrate the effectiveness of the results

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

pinning control of scale free dynamical networks abstract recently it has been demonstrated that many large complex networks display a scale free feature that is their connectivity distributions have the power law form in the present work control of a scale free dynamical network by applying local feedback injections to a fraction of network nodes is investigated the specifically and randomly pinning schemes are considered the specifically pinning of the most highly connected nodes is shown to require a significantly smaller number of local controllers as compared to the randomly pinning scheme the method is applied to an array of chua s oscillators as an example

norm estimates of the cauchy transform and related operators abstract suppose f l p d where p 1 and d is the unit disk let j 0 be the integral operator defined as follows j 0 f z d z 1 w z f w d a w where z w d and d a w 1 d u d v w u i v is the normalized area measure on d suppose j 0 is the adjoint operator of j 0 then j 0 b c where b and c are the operators induced by the bergman projection and cauchy transform respectively in this paper we obtain the l 1 l 2 and l norm of the operator j 0 moreover we obtain the l p d l d norm of the operators c and j 0 provided that p 2 this study is a continuation of the investigations carried out in 4 and 9

internal labels of degenerate representations an expression for the number of internal labels of degenerate irreducible representations of compact semisimple lie groups is given in the same spirit as racah s formula for the nondegenerate case

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

spiral self avoiding walks self avoiding walks with a spiral constraint on the square lattice are enumerated up to 40 steps numerical evidence suggests that they belong to a new universality class

adaptive pinning cluster synchronization of fractional order complex dynamical networks the problem about cluster synchronization of fractional order cdns is studied via a pinning adaptive approach in this paper based on the stability theory of fractional differential equations some sufficient criteria for local and global cluster synchronization of fractional order cdns are derived in this paper the coupling configuration matrix can be asymmetric as well as reducible and the inner coupling matrix can also be asymmetric moreover the number of pinning nodes in each cluster can be evaluated especially when the coupling strength is large enough and the coupling configuration matrix is symmetric cluster synchronization can be achieved via pinning a single node in each cluster finally some typical examples are given to illustrate the correctness and effectiveness of our results a surprising finding is that the synchronization performance will become better as the fractional order decreases in this simulation

canonical orthonormal basis for su 3 contains implies so 3 iii complete set of su 3 tensor operators for pt ii see ibid vol 18 p 1905 1985 a complete set of tensor operators for su 3 is given in a model space generated by two bargmann vectors use is made of modified operator patterns to classify these tensors their tensorial properties are then discussed in the context of the su 3 tensor algebra developed by biedenharn and collaborators 1957 the authors also given analytical semi analytical expressions for important classes of wigner coefficients in gel fand rotational basis

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

critical behaviour of a probabilistic automata network sis model for the spread of an infectious disease in a population of moving individuals a probabilistic automata network sis susceptible infective susceptible model for the spread of an infectious disease in a population of moving individuals is studied the local rule consists of two subrules the first one applied synchronously models infection and recovery it is a probabilistic cellular automaton rule the second applied sequentially describes the motion of the individuals the model contains three parameters the probabilities pi to get infected and pr to recover and the average number of tentative moves per individual m depending upon the values of these parameters in the infinite time limit the system is either in the disease free state or in the endemic state it goes from one state to the other through a transcritical bifurcation similar to a second order phase transition characterized by a non negative order parameter whose role is played in this model by the stationary density of infected individuals the pi pr phase diagram and the critical behaviour of the stationary density of infectives in the neighbourhood of the phase transition are studied as a function of m according to whether the individuals perform short or long range moves it is found that the parameters characterizing the transition have a qualitatively different behaviour as m varies when m is very large the correlations created by the application of the subrule modelling infection and recovery are destroyed and as expected the behaviour of the system is then correctly predicted by a mean field type approximation which assumes a homogeneous mixing of the individuals when m is not large this assumption is no longer correct

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

cluster projective synchronization between community networks with nonidentical nodes abstract in this paper cluster projective synchronization between community networks with nonidentical nodes is investigated outer synchronization between two identical or nonidentical complex networks has been extensively studied in which all the nodes synchronized each other in a common manner however in real community networks different communities in networks usually synchronize with each other in a different manner i e achieving cluster projective synchronization based on lyapunov stability theory sufficient conditions for achieving cluster projective synchronization are derived through designing proper controllers numerical simulations are provided to verify the correctness and effectiveness of the derived theoretical results

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

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

dynamical algebra of the q deformed three dimensional oscillator the q deformed three dimensional harmonic oscillator is defined in terms of the q bosons corresponding to the spherical components of a nondeformed three dimensional oscillator it is shown that the dynamical algebra is spq 6 r two important subalgebra chains are identified spq 6 r suq 3 soq 3 and spq 6 r spq2 2 r soq 3 the basis states of the q deformed oscillator are classified according to these subalgebras finally the hamiltonian eigenvalues are discussed

cluster and group synchronization in delay coupled networks we investigate the stability of synchronized states in delay coupled networks where synchronization takes place in groups of different local dynamics or in cluster states in networks with identical local dynamics using a master stability approach we find that the master stability function shows a discrete rotational symmetry depending on the number of groups the coupling matrices that permit solutions on group or cluster synchronization manifolds show a very similar symmetry in their eigenvalue spectrum which helps to simplify the evaluation of the master stability function our theory allows for the characterization of stability of different patterns of synchronized dynamics in networks with multiple delay times multiple coupling functions but also with multiple kinds of local dynamics in the networks nodes we illustrate our results by calculating stability in the example of delay coupled semiconductor lasers and in a model for neuronal spiking dynamics

first exit time of termites and random super normal conductor networks the resistance of a random super normal conductor network is calculated through the diffusion of a termite which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds the first time t that the termite exits at a distance r from the origin averaged over many configurations and runs is found to behave as t approximately r2k for r xi pc p s describes the divergence of the conductivity as p to pc p being the concentration of the superconductor scaling arguments are used to show that k 1 s 2 theta where 2 theta is the fractal dimension of random walks on single clusters at pc numerical simulations at two dimensions d 2 yields s 1 34 or 0 10 and k 1 34 or 0 03 in agreement with scaling the authors also show that the probability of return to the origin at time t behaves as t dk2 preliminary results at d 1 and other calculations methods are also discussed

scattering operators on fock space ii representations of the scattering operator generated by sl 2 r for pt i see ibid vol 18 p 3141 1985 a representation for unitary scattering operators acting on a symmetric fock space and invariant under an so n internal symmetry group is constructed a group of transformations commuting with so n is seen to be isomorphic to sl 2 r the representations of sl 2 r acting on the fock space are shown to come from the discrete series of representations of sl 2 r these representations are used to label the equivalent irreducible representations of so n and the partial wave amplitudes of the scattering operators are shown to be matrix elements of the discrete series of representations of sl 2 r the example of isospin internal symmetry and the pion triplet is briefly discussed

properties of the density relaxation function in classical diffusion models with percolation transition the relation between the density relaxation function phi and the pair connectedness is shown static and dynamical scaling for phi and quantities related to it are derived from percolation scaling theory due to finite clusters phi contains a non ergodic singularity even in the conducting phase whence a green kubo identity does not hold the form factor of this singularity is discussed for d or 3 the static polarisability can be related to a diverging characteristic length also above the threshold contributions come from confinement in finite clusters and from the structure of the infinite cluster

bloch functions and asymptotic tail variance let p denote the bergman projection on the unit disk dd p z d w 1 zw 2da w z d where dada is normalized area measure we prove that if z 1 z 1 on dd then the integral i a r 02 exp ar4 p rei 2log 11 r2 d 2 0 r 1 has the bound i a r c a 10 1 a 3 2i a r c a 10 1 a 3 2 for 0 1a 1 no such uniform bound is possible we interpret the theorem in terms the asymptotic tail variance of such a bergman projection p p by the way the asymptotic tail variance induces a seminorm on the bloch space this improves upon earlier work of makarov which covers the range 0 a 264 0 1542 we then apply the theorem to obtain an estimate of the universal integral means spectrum for conformal mappings with a k quasiconformal extension for 0 k 10 k 1 the estimate reads for t ct c and 0 k 10 k 1 b k t 14k2 t 2 1 7k 2 for t 2k 1 7k 2 k t 1 1 7k 2 for t 2k 1 7k 2 which should be compared with the conjecture by prause and smirnov to the effect that for real t with t 2 k t 2 k we should have b k t 14k2t2

covariance correlation and entanglement some new identities for quantum variance and covariance involving commutators are presented in which the density matrix and the operators are treated symmetrically a measure of entanglement is proposed for bipartite systems based upon covariance this works for two and three component systems but produces ambiguities for multicomponent systems of composite dimension its relationship to angular momentum dispersion for symmetric spin states is described

sharpening holder s inequality abstract we strengthen holder s inequality the new family of sharp inequalities we obtain might be thought of as an analog of the pythagorean theorem for the l p spaces our treatment of the subject matter is based on bellman functions of four variables

tests of hyperuniversality for self avoiding walks a universal combination of amplitudes for the end to end distance of n step self avoiding walks and for the number of n step self avoiding polygons is estimated numerically for various lattices in d 2 universality of this amplitude combination is confirmed to good accuracy while in d 3 the data are consistent with universality but the error limits are rather large for finite width square lattice strips the authors calculate universal critical point correlation amplitudes by fixed fugacity monte carlo simulations in addition these amplitudes are calculated analytically either exactly or approximately by cardy s conformal mapping approach numerical values agree well with the theoretical predictions

kronecker product of sp 2n representations using generalised young tableaux using generalised young tableaux the authors obtain an explicit formula for the reduction of the kronecker product of irreducible representations of the symplectic groups this extends a previous work devoted to the case of orthogonal groups

canonical solution of the state labelling problem for su n so n and littlewood s branching rule ii use of modification rules for pt i see ibid vol 16 p 2095 1983 in part i it was shown that when d 1 2n case where littlewood s branching rule 1950 for u n contains implies o n has to be supplemented with newell s modification rules 1951 the explicit solution of the state labelling problem for v 1 row irreducible representations of both su 2v 1 and su 2v is given and is shown to reflect the operation of littlewood s modified branching rule in a direct way the generalisation to higher d values is then outlined

a new algorithm to enumerate the self avoiding random walk presents a new algorithm for enumerating the self avoiding walk and extend the existing series of 19 terms by one more term on a triangular lattice the leading correction to scaling exponent delta 1 for the mean square end to end distance of the self avoiding walk is found to be less than one

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

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

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

a hopf laboratory for symmetric functions an analysis of symmetric function theory is given from the perspective of the underlying hopf and bi algebraic structures these are presented explicitly in terms of standard symmetric function operations particular attention is focused on laplace pairing sweedler cohomology for 1 and 2 cochains and twisted products rota cliffordizations induced by branching operators in the symmetric function context the latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type a commentary on related issues in the combinatorial approach to quantum field theory is given

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found

on the apparent failure of a flory approximation for directed linear polymers the author discusses the apparent failure of a flory approximation for directed linear polymers commenting on the details implicit in this type of picture the relationship of a standard flory viewpoint to a simplified application of fractal theory is explored and a correspondence is drawn between aspects of both theories