on resolving the multiplicity of arbitrary tensor products of the u n groups representations of u n are realised as right translations on holomorphic hilbert bargmann spaces of n n complex variables r fold tensor product spaces of irreducible representations of u n are shown to be isomorphic to subspaces of the holomorphic hilbert spaces maps are exhibited which carry an irreducible representation of u n into these subspaces the algebra of operators commuting with these maps is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity several examples from u 3 are explicitly worked out

vector coherent state constructions of u 3 symmetric tensors and their su 3 su 2 u 1 wigner coefficients generalized vector coherent state constructions of totally symmetric u 3 tensors are used to gain new expressions for the su 3 su 2 u 1 wigner coefficients for the coupling 1 1 20 3 3 these expressions show how the extremely simple formulas of le blanc and biedenharn involving a single 9 j coefficient arise as special cases of a general result that involves 12 j coefficients a simpler general result involving only 9 j coefficients and k normalization factors is derived in a way that can in principle be generalized to the generic coupling with multiplicity

random attractors for stochastic porous media equations perturbed by space time linear multiplicative noise unique existence of solutions to porous media equations driven by continuous linear multiplicative space time rough signals is proven for initial data in l 1 o on bounded domains o the generation of a continuous order preserving random dynamical system rds on l 1 o and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained the random attractor is shown to be compact and attracting in l o norm uniform l bounds and uniform space time continuity of the solutions is shown general noise including fractional brownian motion for all hurst parameters is treated a pathwise wong zakai result for driving noise given by a continuous semimartingale is obtained for fast diffusion equations driven by continuous linear multiplicative space time rough signals existence of solutions is proven for initial data in l m 1 o

random attractors for quasi continuous random dynamical systems and applications to stochastic reaction diffusion equations abstract this paper first introduces the so called quasi continuous random dynamical system rds on a separable banach space the quasi continuity is weaker than all the usual continuities and thus is easier to check in practice we then establish a necessary and sufficient condition for the existence of random attractors for the quasi continuous rds we also give a general method to obtain the random attractors for the rds on the banach space l q d for q 2 as an application it is shown that the rds generated by the stochastic reaction diffusion equation possesses a finite dimensional random attractor in l q d for any q 2 a comparison result of fractal dimensions under the different l q norms is also obtained

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

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

the asymptotic behavior of the stochastic ginzburg landau equation with additive noise abstract we show that the stochastic ginzburg landau equation with additive noise can be solved pathwise and the unique solution generates a random dynamic system then we prove that the system possesses a compact random attractor in l 2 d when the spatial dimension of d is one and two respectively

subduction coefficients of brauer algebras and racah coefficients of o n and sp 2 m ii racah coefficients irreducible representations of brauer algebras are discussed in the non standard basis a method for evaluating subduction coefficients sdcs i e the transformation coefficients between standard and non standard bases of brauer algebras is outlined non trivial sdcs of for are derived racah coefficients of o n and sp 2m can be derived from subduction coefficients of brauer algebras by using the schur weyl duality relation between and o n or sp 2m

random attractors for stochastic 2d navier stokes equations in some unbounded domains abstract we show that the stochastic flow generated by the 2 dimensional stochastic navier stokes equations with rough noise on a poincare like domain has a unique random attractor one of the technical problems associated with the rough noise is overcomed by the use of the corresponding cameron martin or reproducing kernel hilbert space our results complement the result by brze niak and li 2006 10 who showed that the corresponding flow is asymptotically compact and also generalize caraballo et al 2006 12 who proved existence of a unique attractor for the time dependent deterministic navier stokes equations

pullback attractors in v for non autonomous 2d navier stokes equations and their tempered behaviour abstract in this paper the asymptotic behaviour of the solutions to a non autonomous 2d navier stokes model is analyzed when the initial datum belongs to v for two frameworks the universe of fixed bounded sets and also for another universe given by a tempered condition the existence of pullback attractors in these different universes is established and thanks to regularity properties the relation between these several families of attractors and the corresponding in h is successfully studied finally two results about the tempered behaviour in v and h 2 2 of the pullback attractors when time goes to are obtained

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

generalized vector coherent state theory of sp 2n r vector operators and of sp 2n r contains implies u n reduced wigner coefficients sp 2n r contains implies u n reduced wigner coefficients coupling a positive discrete series unitary irreducible representation with the 2n dimensional non unitary irreducible representation characterized by 1 are evaluated by making use of generalized vector coherent state techniques general formulae are given in terms of the k matrix elements of vector coherent state theory and of some u n 6j or 9j type recoupling coefficients explicit analytic expressions are obtained in some important special cases such results should allow a determination of wsp 2n r approximately w n p 2n r and osp p 2n r matrix representations to be completed because both the w n generators and the osp p 2n r odd generators transform under the sp 2n r irrep 1

highest weight representations for gl m n and gl m n it is shown how vector coherent state vcs theory enlightens the simultaneous discussion of representation theory for classical lie algebras and superalgebras and provides an optimal framework for the explanation of the noted similarities and dissimilarities of their representations reducibility atypicality and the positive definitiveness of inner products in vcs representation spaces are discussed the discussion is exemplified through the parallel and explicit construction of highest weight ladder representations of the lie algebra gl m n and superalgebra gl m n in gl m gl n bases

upper semi continuity and regularity of random attractors on p times integrable spaces and applications abstract a sufficient condition for a family of random attractors to be upper semi continuous and regular is obtained when the state space is a p times integrable space with p 2 it is shown that both upper semi continuity and regularity under the topology of p norms do not rely on continuity convergence and absorption of the associated random dynamical systems in l p since the corresponding l 2 properties can offer all as an application of the abstract result it is shown that the family of random attractors for the stochastic reaction diffusion equations on a unbounded domain is upper semi continuous and regular at any point under the topology of p norms

the existence of global attractors for the norm to weak continuous semigroup and application to the nonlinear reaction diffusion equations abstract in this paper first we introduce a new concept called the norm to weak continuous semigroup in a banach space and give a technical theorem to verify this notion of continuity then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup as an application we obtain the existence of the global attractor for a nonlinear reaction diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term the global attractors are obtained in l p h 0 1 and h 2 h 0 1 respectively a new a priori estimate method called asymptotic a priori estimate has been introduced since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in l p h 0 1 and h 2 h 0 1 the results in this part are new and appear to be optimal

existence and upper semicontinuity of bi spatial pullback attractors for smoothing cocycles abstract in this work we establish several criteria for the existence as well as the upper semi continuity of bi spatial attractors under a closedness condition which dramatically weakens the usual requirement on the continuity of the cocycle it is also shown that though the continuity plays a less important role in the study of attractors it is impossible to establish an existence criteria for common attractors for systems without any continuity like properties however for such bad systems one can expect a mini attractor which is shown adequate well to depict the asymptotic behavior of non continuous systems finally we study the l 2 h 0 1 pullback attractor for a stochastic complex ginzburg landau equation a spectrum decomposition method is employed to overcome the lack of sobolev compactness embeddings in h 0 1

lapped unimodular transforms lifting factorization and structural regularity in this paper the lifting factorization and structural regularity of the lapped unimodular transforms luts are studied the proposed m channel lifting factorization is complete is minimal in the mcmillan sense and has diagonal entries of unity in addition to allowing for integer to integer mapping and guaranteeing perfect reconstruction even under finite precision the proposed lifting factorization structurally ensures unimodularity for regular lut design structural conditions that impose 1 1 1 2 and 2 1 regularity onto the filter banks fbs are presented consequently the optimal filter coefficients can be obtained through unconstrained optimizations a special lifting based lattice structure is used for parameterizing nonsingular matrices which not only helps impose regularity but also has rational coefficient unimodular fbs as a by product the regular luts can be transformed to the lifting domain with the proposed factorization for faster and multiplierless implementations the lifting factorization and the regularity conditions are derived for two different type i and type ii factorizations of the first order unimodular fbs design examples are presented to confirm the proposed theory

on some racah coefficients of u n some u n racah coefficients appearing in various applications of vector coherent state and k matrix theories are calculated for such a purpose use is made of their definition in terms of u n u n 1 reduced wigner coefficients and u n 1 racah coefficients by starting from known u 2 racah coefficients the recursion relations obtained are solved by induction over n

on resolving the multiplicity of tensor products of irreducible representations of symplectic groups the multiplicity problem of r fold tensor products of irreducible representations of sp 2k c are considered the arbitrary r fold tensor product is shown to be isomorphic to a subspace of the holomorphic hilbert bargmann space of n 2k complex variables maps are constructed which carry an irreducible representation of sp 2k c into this subspace an algebra of commuting operators is constructed and it is shown how eigenvalues and eigenvectors of certain of these operators can be used to resolve the multiplicity

lapped unimodular transform and its factorization two types of lapped transforms have been studied in detail in the literature namely the lapped orthogonal transform lot and its extension the biorthogonal lapped transform bolt in this paper we will study the lapped unimodular transform lut all three transforms are first order matrices with finite impulse response fir inverses we will show that like lot and bolt all luts can be factorized into degree one unimodular matrices the factorization is both minimal and complete we will also show that all first order systems with fir inverses can be minimally factorized as a cascade of degree one lot bolt and lut building blocks two examples will be given to demonstrate that despite having a very small system delay the luts have a satisfactory performance in comparison with lot and bolt

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

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

every unit matrix is a lulu abstract the four matrices l0u0l1u1 at the end of the title are triangular with ones on their main diagonals their product has determinant one following a question and theorem of toffoli we show that any matrix with determinant one can be factored in this way a transformation of the plane becomes a sequence of one dimensional shears with n2 1 free parameters

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

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

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

limiting dynamics for stochastic reaction diffusion equations on the sobolev space with thin domains abstract this paper is devoted to bi spatial random attractors of the stochastic reaction diffusion equation when the terminate space is the sobolev space on a thin domain where the nonlinearity can be decomposed into two functions with p q growth exponents by means of a computation method of induction it is shown that the difference of solutions near the initial time is integrable of k p 2 k 2 order this higher order integrability shows continuity of the solution operator from the square lebesgue space to k p 2 k 2 order lebesgue spaces in particular the 2 p 2 order integrability shows continuity of the solution operator from the square lebesgue space to the sobolev space which further shows existence of a random attractor in the sobolev space when the initial space is the square lebesgue space moreover the higher order integrability can be uniform with respect to all thin domains which provides uniformly asymptotic compactness of the random dynamical systems as a conclusion the upper semi continuity of attractors under the sobolev norm is established when the narrow domain degenerates onto a lower dimensional domain

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

implementation of the u n tensor operator calculus in a vector bargmann hilbert space using a vector coherent state framework and an associated vector bargmann hilbert space 1961 the structure of two classes of u n u n 1 unit projective operators is shown to be intimately related to 6 j and 9 j coefficients of the u n 1 subalgebra explicit verification of limit properties for these operators allows the unambiguous assignment of a set of canonical operator labels

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

flattening squeezing and the existence of random attractors the study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems in the deterministic case the properties of flattening and squeezing in infinite dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor the flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories it is shown here that the flattening property is implied by the squeezing property and is in fact weaker since the attractor in a system with the flattening property can be infinite dimensional whereas it is always finite dimensional in a system with the squeezing property the flattening property is then generalized to random dynamical systems for which it is called the pullback flattening property it is shown to be weaker than the random squeezing property but equivalent to pullback asymptotic compactness and pullback limit set compactness and thus implies the existence of a random attractor the results are also valid for deterministic non autonomous dynamical systems formulated as skew product flows

regularity and structure of pullback attractors for reaction diffusion type systems without uniqueness abstract in this paper we study the pullback attractor for a general reaction diffusion system for which the uniqueness of solutions is not assumed we first establish some general results for a multi valued dynamical system to have a bi spatial pullback attractor and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories as an application a general reaction diffusion system is proved to have an invariant h v pullback attractor a a r this attractor is composed of all the backwards compact complete trajectories of the system pullback attracts bounded subsets of h in the topology of v and moreover s a s is precompact in v r a non autonomous fitz hugh nagumo equation is studied as a specific example of the reaction diffusion system

upper semicontinuity of attractors of stochastic delay reaction diffusion equations in the delay a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero a system of stochastic delayed reaction diffusion equations with multiplicative noise and deterministic non autonomous forcing is considered we first prove the existence and uniqueness of a bi spatial pullback attractor for these equations when the initial space is c 0 l2o and the terminate space is c 0 h01o the asymptotic compactness of solutions in c 0 h01o is established by combining positive and negative truncations technique and some new estimates on solutions then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions finally upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero

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

theory and factorization for a class of structurally regular biorthogonal filter banks regularity is a fundamental and desirable property of wavelets and perfect reconstruction filter banks prfbs among others it dictates the smoothness of the wavelet basis and the rate of decay of the wavelet coefficients this paper considers how regularity of a desired degree can be structurally imposed onto biorthogonal filter banks bofbs so that they can be designed with exact regularity and fast convergence via unconstrained optimization the considered design space is a useful class of m channel causal finite impulse response fir bofbs having anticausal fir inverses that are characterized by the dyadic based structure w z i uv sup spl dagger z sup 1 uv sup spl dagger for which u and v are m spl times spl gamma parameter matrices satisfying v sup spl dagger u i sub spl gamma 1 spl les spl gamma spl les m for any m spl ges 2 structural conditions for regularity are derived where the householder transform is found convenient as a special case a class of regular linear phase bofbs is considered by further imposing linear phase lp on the dyadic based structure in this way an alternative and simplified parameterization of the biorthogonal linear phase filter banks glbts is obtained and the general theory of structural regularity is shown to simplify significantly regular bofbs are designed according to the proposed theory and are evaluated using a transform based image codec they are found to provide better objective performance and improved perceptual quality of the decompressed images specifically the blocking artifacts are reduced and texture details are better preserved for fingerprint images the proposed biorthogonal transform codec outperforms the fbi scheme by 1 1 6 db in psnr

role of anticausal inverses in multirate filter banks ii the fir case factorizations and biorthogonal lapped transforms for pt i see ibid vol 43 no 5 p 1090 1990 in part i we studied the system theoretic properties of discrete time transfer matrices in the context of inversion and classified them according to the types of inverses they had in particular we outlined the role of causal fir matrices with anticausal fir inverses abbreviated cafacafi in the characterization of fir perfect reconstruction pr filter banks essentially all fir pr filter banks can be characterized by causal fir polyphase matrices having anticausal fir inverses in this paper we introduce the most general degree one cafacafi building block and consider the problem of factorizing cafacafi systems into these building blocks factorizability conditions are developed a special class of cafacafi systems called the biorthogonal lapped transform bolt is developed and shown to be factorizable this is a generalization of the well known lapped orthogonal transform lot examples of unfactorizable cafacafi systems are also demonstrated finally it is shown that any causal fir matrix with fir inverse can be written as a product of a factorizable cafacafi system and a unimodular matrix

new perspective on the u n wigner racah calculus ii elementary reduced wigner coefficients for u n for pt i see ibid vol 20 no 9 p 2241 50 1987 exploiting the powerful formalism of the vector coherent state theory expounded in part i the group theoretical content of the complementarity principle and a novel interpretation of the operator pattern of biedenharn and louck the authors rederive in a simple fashion all u n u n 1 reduced wigner coefficients for the elementary u n tensors

the matrix representations of g2 i representations in an so 4 basis irreducible representations of the real compact lie algebra g2 are given in g2 so 4 bases all missing labels are accounted for by the explicit construction of a g2 so 4 basis of vector holomorphic functions the general problem of missing internal labels is also briefly 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

a resolution of the su 3 outer multiplicity problem and computation of wigner coefficients for su 3 a simple algorithm is given for the resolution of the su 3 multiplicity problem and the computation of su 3 wigner coefficients using a complete set of u 2 x u 3 bargmann tensors classified by operator patterns null space properties of these tensors are easily derived their structure is such that a direct one to one correspondence is shown to exist between the bargmann tensors and the terms in the clebsch gordan series for su 3 as derived by o reilly 1982 finally the resolution presented herein is shown to be concordant with an alternative resolution advocated long ago by hecht

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

vector coherent state theory of the semidirect sum lie algebras wsp 2n r positive discrete series irreducible representations of the semidirect sum lie algebras wsp 2n r w n sp 2n r are given in wsp 2n r contains implies sp 2n r contains implies u n bases the branching rule for the reduction of a wsp 2n r irreducible representation into a direct sum of multiplicity free sp 2n r irreducible representations is accounted for by the explicit construction of a wsp 2n r contains implies sp 2n r contains implies u n basis of vector valued holomorphic functions

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

a class of regular biorthogonal linear phase filterbanks theory structure and application in image coding this paper discusses a method of regularity imposition onto biorthogonal linear phase m band filterbanks using the lattice structure a lifting structure is proposed for lattice matrix parameterization where regularity constraints can be imposed the paper focuses on cases with analysis and synthesis filterbanks having up to two degrees of regularity necessary and sufficient conditions for regular filterbanks in terms of the filter impulse response frequency response scaling function and wavelets are revisited and are derived in terms of the lattice matrices this also leads to a constraint on the minimum filter length presented design examples are optimized for the purpose of image coding i e the main objectives are coding gain and frequency selectivity simulation results from an image coding application also show that these transforms yield improvement in the perceptual quality in the reconstruction images the approach has also been extended to the case of integer rational lifting coefficients which are desirable in many practical applications

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

sufficient and necessary criteria for existence of pullback attractors for non compact random dynamical systems abstract we study pullback attractors of non autonomous non compact dynamical systems generated by differential equations with non autonomous deterministic as well as stochastic forcing terms we first introduce the concepts of pullback attractors and asymptotic compactness for such systems we then prove a sufficient and necessary condition for existence of pullback attractors we also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors for random systems containing periodic deterministic forcing terms we show the pullback attractors are also periodic under certain conditions as an application of the abstract theory we prove the existence of a unique pullback attractor for reaction diffusion equations on r n with both deterministic and random external terms since sobolev embeddings are not compact on unbounded domains the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions

screening and vertex operators for u n screening and vertex operators for u n are given in terms of generators of the group left action on the flag manifold u n 1 su n their matrix elements in a basis of holomorphic sections of a vector bundle are computed

the matrix representations of g2 ii representations in an su 3 basis irreducible representations of the real compact lie algebra g2 are given in g2 su 3 bases a missing label is accounted for by the explicit construction of a g2 su 3 basis of vector holomorphic functions analytical results are given for two multiplicity free classes of irreps it is also shown how vector coherent state vcs theory accommodates the decomposition of the nilpotent raising operator subalgebra of an arbitrary lie algebra into a finite but arbitrary number of irreducible tensorial sets under transformations generated by a stability algebra

dyadic based factorizations for regular paraunitary filterbanks and m band orthogonal wavelets with structural vanishing moments paraunitary filterbanks pufbs can be designed and implemented using either degree one or order one dyadic based factorization this work discusses how regularity of a desired degree is structurally imposed on such factorizations for any number of channels m spl ges 2 without necessarily constraining the phase responses the regular linear phase pufbs become a special case under the proposed framework we show that the regularity conditions are conveniently expressed in terms of recently reported m channel lifting structures which allow for fast reversible and possibly multiplierless implementations in addition to improved design efficiency as suggested by numerical experience m band orthonormal wavelets with structural vanishing moments are obtained by iterating the resulting regular pufbs on the lowpass channel design examples are presented and evaluated using a transform based image coder and they are found to outperform previously reported designs

global attractors for the p laplacian equations with nonregular data abstract let be a smooth bounded domain in r n n 3 we consider the long time behavior of solutions to the p laplacian equation u t p u f u g where 2 p n p div u p 2 u is the p laplace operator assume that g and the initial condition u 0 lie in l 1 f r r is of class c 1 and satisfies proper growth conditions firstly we prove the uniqueness of the entropy solution and establish some regularity results then we show the existence of a global attractor a in l r 1 w 0 1 s with s max n p 1 n 1 p r 1 r to obtain the results a decomposition method and a bootstrap technique are used

existence and continuity of bi spatial random attractors and application to stochastic semilinear laplacian equations abstract a concept of a bi spatial random attractor for a random dynamical system is introduced a unified result about existence and upper semi continuity for a family of bi spatial random attractors is obtained if a family of random systems is convergent uniformly absorbing in an initial space and uniformly omega compact in both initial and terminate spaces the upper semi continuity result improves all existing results even for single spatial attractors as an application of the abstract result it is shown that every semilinear laplacian equation on the entire space perturbed by a multiplicative and stochastic noise possesses an l 2 l q random attractor with q 2 moreover it is proved that the family of obtained attractors is upper semi continuous at any density of noises and the family of attractors for the corresponding compact systems is both upper and lower semi continuous at infinity under the topology of both spaces

on the symmetry properties of outer product reduction the symmetry properties of outer product coupling coefficients for the symmetric group and of clebsch gordan coefficients for the unitary group and of the corresponding isoscalar factors are discussed a simple rule is given for determination of a phase factor of outer reduction symmetry of sn and u n the rule is provided with a relationship between the phase factor of a young diagram with its subdiagrams

new perspective on the u n wigner racah calculus i vector coherent state theory and construction of gel fand bases using a vector coherent state theory it is shown that the construction of gel fand bases for the unitary group is particularly simple the very specific rules for construction of the states greatly facilitate the subsequent computation of matrix elements of the generators of the u n lie algebra

asymptotic smoothing and the global attractor of a weakly damped kdv equation on the real line abstract the existence of the global attractor of a weakly damped forced korteweg de vries equation in the phase space l 2 r is proved an optimal asymptotic smoothing effect of the equation is also shown namely that for forces in l 2 r the global attractor in the phase space l 2 r is actually a compact set in h 3 r the energy equation method is used in conjunction with a suitable splitting of the solutions the dispersive regularization properties of the equation in the context of bourgain spaces are extensively exploited

complementary group resolution of the su n outer multiplicity problem i the littlewood rules and a complementary u 2n 2 group structure a complementary group to su n is found that realizes all features of the littlewood rules for kronecker products of su n representations this is accomplished by considering a state of su n to be a special gel fand state of the complementary group u 2n 2 with labels of the latter used to distinguish multiple occurrences of irreducible representations of su n irreps in the su n su n su n decomposition that is obtained from the littlewood rules furthermore this realization also helps us to determine su n su n 1 u 1 reduced wigner coefficients rwcs frequently called isoscalar factors and clebsch gordan coefficients cgcs or full nonreduced wigner coefficients of su n using algebraic or numeric methods in either the canonical or a noncanonical basis new explicit formulas for the su 3 and su 4 multiplicities are obtained by using this technique

uniform attractors for non autonomous random dynamical systems abstract this paper is devoted to establishing a random uniform attractor theory for non autonomous random dynamical systems nrds the uniform attractor is defined as the minimal compact uniformly pullback attracting random set nevertheless the uniform pullback attraction in fact implies a uniform forward attraction in probability and implies also an almost uniform pullback attraction for discrete time sequences though no invariance is required by definition the uniform attractor can have a negative semi invariance under certain conditions several existence criteria for uniform attractors are given and the relationship between uniform and cocycle attractors is carefully studied to overcome the measurability difficulty the symbol space is required to be polish which is shown fulfilled by the hulls of l l o c p r l r functions p r 1 moreover uniform attractors for continuous nrds are shown determined by uniformly attracting deterministic compact sets finally the uniform attractor for a stochastic reaction diffusion equation with translation bounded external forcing are studied as applications

fast multiplierless approximations of the dct with the lifting scheme we present the design implementation and application of several families of fast multiplierless approximations of the discrete cosine transform dct with the lifting scheme called the bindct these bindct families are derived from chen s 1977 and loeffler s 1989 plane rotation based factorizations of the dct matrix respectively and the design approach can also be applied to a dct of arbitrary size two design approaches are presented in the first method an optimization program is defined and the multiplierless transform is obtained by approximating its solution with dyadic values in the second method a general lifting based scaled dct structure is obtained and the analytical values of all lifting parameters are derived enabling dyadic approximations with different accuracies therefore the bindct can be tuned to cover the gap between the walsh hadamard transform and the dct the corresponding two dimensional 2 d bindct allows a 16 bit implementation enables lossless compression and maintains satisfactory compatibility with the floating point dct the performance of the bindct in jpeg h 263 and lossless compression is also demonstrated

vector coherent state theory and its application to the orthogonal groups vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple lie groups and their lie algebras it is shown that in a suitable basis the vector coherent state inner product can be inferred algebraically by k matrix theory and changed to a simpler bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of lie algebras applications are made to the even and odd orthogonal lie algebras

stability of random attractors under perturbation and approximation abstract the comparison of the long time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals however one can still compare their asymptotic behaviour using the global attractor and this is now standard in the deterministic autonomous case for random dynamical systems there is an additional problem since the convergence of numerical methods for such systems is usually given only on average in this paper the deterministic approach is extended to cover stochastic differential equations giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi continuous with respect to a given family of perturbations or approximations

integer fast fourier transform a concept of integer fast fourier transform intfft for approximating the discrete fourier transform is introduced unlike the fixed point fast fourier transform fxpfft the new transform has the properties that it is an integer to integer mapping is power adaptable and is reversible the lifting scheme is used to approximate complex multiplications appearing in the fft lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations split radix fft is used to illustrate the approach for the case of 2 sup n point fft in which case an upper bound of the minimal dynamic range of the internal nodes which is required by the reversibility of the transform is presented and confirmed by a simulation the transform can be implemented by using only bit shifts and additions but no multiplication a method for minimizing the number of additions required is presented while preserving the reversibility the intfft is shown experimentally to yield the same accuracy as the fxpfft when their coefficients are quantized to a certain number of bits complexity of the intfft is shown to be much lower than that of the fxpfft in terms of the numbers of additions and shifts finally they are applied to noise reduction applications where the intfft provides significantly improvement over the fxpfft at low power and maintains similar results at high power

complete factorization for poly phase matrix with linear phase based on semi rank orthogonal projection matrix abstract this paper puts forward a new factorization named the semi rank factorization for para unitary poly phase matrix with linear phase and reveals the equivalent relationship between the orthogonal projection matrix and the linear phase and characterizes the orthogonal projection matrix the semi rank factorization has the completeness property no matter for even or odd number of channels in addition a decomposition technique for the starting block matrix is developed using the reflection matrix the semi rank factorization provides an efficient and compact representation for multi band filter banks with linear phase in particular the complete representation for multi band symmetry filter banks with odd number of channels is brand new as a result the programmable algorithms are obtained for designs of optimal filter banks based on the complete representation the optimal examples are presented to illustrate the excellent performance on image compression

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

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

relationship between sn and u n isoscalar factors and higher order u n invariants general relationships expressing u n and sn coupling and transformation isoscalar factors in terms of u n racah and 9 coefficients are derived the absolute values of u n racah coefficients involving at most k column irreducible representations are shown to be identical with su k racah coefficients in particular an explicit relationship is established between the u n and su 2 approaches to the many electron correlation problem

subduction coefficients of birman wenzl algebras and racah coefficients of the quantum groups oq n and spq 2m i subduction coefficients irreducible representations of birman wenzl algebras cf r q in the non standard basis are discussed a procedure for evaluating subduction coefficients sdcs or the transformation coefficients between standard and non standard basis of birman wenzl algebras is formulated based on the linear equation method sdcs of cf r q with f 4 are derived racah coefficients of the quantum groups oq n and spq 2m can be obtained from sdcs of birman wenzl algebras cf r q by using the schur weyl brauer duality relation between birman wenzl algebras cf r q and the quantum groups oq n and spq 2m

induced shift tensors in vector coherent state theory it is shown that the so called shift tensors of biedenharn and louck have a natural expression within the framework of vector coherent state vcs theory vcs theory was developed to give a systematic construction of the representations of lie groups and to derive the corresponding matrices representing the elements of their lie algebras in this paper there are explicit realizations of shift tensors within the framework of vcs theory for semisimple lie groups in terms of shift tensors for suitably defined subgroups

the asymptotic behavior of the stochastic ginzburg landau equation with multiplicative noise the asymptotic behavior of the stochastic ginzburg landau equation is studied we obtain the stochastic ginzburg landau equation as a finite dimensional random attractor

new perspective on the u n wigner racah calculus iii applications to u 2 and u 3 for pt ii see ibid vol 20 no 14 p 4613 35 1987 using the framework introduced in parts i and ii the author derives an interesting relationship between u 3 symmetric tensors and u 2 racah coefficients closed formulae for generic u 2 wigner and racah coefficients are given in terms of alternative parametrisations for such coefficients