baker campbell hausdorff relation for special unitary groups su n multiplication of two elements of the special unitary group su n determines uniquely a third group element a baker campbell hausdorff relation is derived which expresses the group parameters of the product written as an exponential in terms of the parameters of the exponential factors this requires the eigenvalues of three matrices consequently the relation can be stated analytically up to n 4 in principle similarity transformations encoding the time evolution of quantum mechanical observables for example can be obtained by the same means

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

decentralized two filter smoothing algorithms in discrete time systems we develop some decentralized fixed interval smoothing algorithms which are realized by the two filter form in linear discrete time systems where the decentralized estimation structure consists of a central processor and of two local processors two approaches based on local filtering and local smoothing are considered for solving the problems of decentralized smoothing smoothing update and real time smoothing it is then shown that the notion of decentralized kalman filtering plays a significant role in the derivation of the algorithms for the former approach and that the introduction of the hamiltonian system in discrete time immediately gives the solutions of the problems in the latter approach moreover some characteristics of the algorithms arising from each approach are also discussed

coherent states for the unitary symplectic group we present an explicit construction of coherent states for an arbitrary irreducible representation of the unitary symplectic group usp 4 three different families of coherent states are obtained corresponding to the subgroups u 1 u 1 u 2 and su 2 su 2 the symplectic structure on the manifold of coherent states is obtained and canonical coordinates are used to express the classical limit of quantum observables one of the families is seen to provide a trivial classical limit

strong and weak error estimates for elliptic partial differential equations with random coefficients we consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous dirichlet boundary conditions we focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness this model is frequently used in hydrogeology we approximate this coefficient by a finite dimensional noise using a truncated karhunen loeve expansion we give estimates of the corresponding error on the solution both a strong error estimate and a weak error estimate that is an estimate of the error commited on the law of the solution we obtain a weak rate of convergence which is twice the strong one in addition we give a complete error estimate for the stochastic collocation method in this case where neither coercivity nor boundedness is stochastically uniform to conclude we apply these results of strong and weak convergence to two classical cases of covariance kernel choices the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel yielding explicit weak and strong convergence rates

square root rts smoothing algorithms two kinds of new square root algorithms for rauch tung striebel rts smoothing formulas are suggested stable square root algorithms and a fast square root or so called chandrasekhar type algori

on biunimodular vectors for unitary matrices abstract a biunimodular vector of a unitary matrix a u n is a vector v t n c n such that av t n as well over the last 30 years the sets of biunimodular vectors for fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences here we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices this search can be motivated in various ways the main motivation is provided by the fact that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices

prepotential formulation of su 3 lattice gauge theory the su 3 lattice gauge theory is reformulated in terms of su 3 prepotential harmonic oscillators this reformulation has enlarged su 3 u 1 u 1 gauge invariance under which the prepotential operators transform like matter fields the hilbert space of su 3 lattice gauge theory is shown to be equivalent to the hilbert space of the prepotential formulation satisfying certain color invariant sp 2 r constraints the su 3 irreducible prepotential operators which solve these sp 2 r constraints are used to construct su 3 gauge invariant hilbert spaces at every lattice site in terms of su 3 gauge invariant vertex operators the electric fields and the link operators are reconstructed in terms of these su 3 irreducible prepotential operators we show that all the su 3 mandelstam constraints become local and take a very simple form within this approach we also discuss the construction of all possible linearly independent su 3 loop states which solve the mandelstam constraints the techniques can be easily generalized to su n

probabilistic equivalence and stochastic model reduction in multiscale analysis this paper presents a probabilistic upscaling of mechanics models a reduced order probabilistic model is constructed as a coarse scale representation of a specified fine scale model whose probabilistic structure can be accurately determined equivalence of the fine and coarse scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained a significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine and coarse scale representations the upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations an illustration in stochastic structural dynamics is provided to demonstrate the proposed framework

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

new square root smoothing algorithms this paper presents new square root smoothing algorithms for the three best known smoothing formulas 1 rauch tung striebel rts formulas 2 desai weinert yusypchuk dwy formulas called backward rts formulas and 3 mayne fraser mf formulas called two filter formulas the main feature of the new algorithms is that they use unitary rotations to replace all matrix inversion and backsubstitution steps common in earlier algorithms with unitary operations this feature enables more efficient systolic array and parallel implementations and leads to algorithms with better numerical stability and conditioning properties

an algebraic algorithm for calculating clebsch gordan coefficients application to su 2 and su 3 a recent paper gave an explicit construction for inducing shift tensors of a compact reductive lie group from shift tensors of a suitably defined subgroup the shift tensors were defined on model spaces of holomorphic vector coherent state wave functions in this paper we use these shift tensors to obtain an algorithm for computing clebsch gordan coefficients the approach reproduces the known analytical results for su 2 and gives a simple algorithm for computing su 3 coefficients the algorithm is shown to yield analytical expressions for the multiplicity free su 3 couplings of type 20 10

cluster based generalized multiscale finite element method for elliptic pdes with random coefficients abstract we propose a generalized multiscale finite element method gmsfem based on clustering algorithm to study the elliptic pdes with random coefficients in the multi query setting our method consists of offline and online stages in the offline stage we construct a small number of reduced basis functions within each coarse grid block which can then be used to approximate the multiscale finite element basis functions in addition we coarsen the corresponding random space through a clustering algorithm in the online stage we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre computed multiscale basis the new gmsfem can be applied to multiscale spde starting with a relatively coarse grid without requiring the coarsest grid to resolve the smallest scale of the solution the new method offers considerable savings in solving multiscale spdes numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation

generalized coherent states for su n systems generalized coherent states are developed for su n systems for arbitrary n this is done by first iteratively determining explicit representations for the su n coherent states and then determining parametric representations useful for applications for su n the set of coherent states is isomorphic to a coset space su n su n 1 and thus shows the geometrical structure of the coset space these results provide a convenient 2n 1 dimensional space for the description of arbitrary su n systems we further obtain the metric and measure on the coset space and show some properties of the su n coherent states

analytic representations based on su 1 1 coherent states and their applications we consider two analytic representations of the su 1 1 lie group the representation in the unit disc based on the su 1 1 perelomov coherent states and the barut girardello representation based on the eigenstates of the su 1 1 lowering generator we show that these representations are related through a laplace transform a weak resolution of the identity in terms of the perelomov su 1 1 coherent states is presented which is valid even when the bargmann index k is smaller than various applications of these results in the context of the two photon realization of su 1 1 in quantum optics are also discussed

su n coherent states and irreducible schwinger bosons we exploit the su n irreducible schwinger boson to construct su n coherent states this construction of an su n coherent state is analogous to the construction of the simplest heisenberg weyl coherent states the coherent states belonging to irreducible representations of su n are labeled by the eigenvalues of n 1 su n casimir operators and characterized by n 1 complex orthonormal vectors describing the su n group manifold

paper a new computationally efficient fixed interval discrete time smoother the rauch tung streibel smoother recursion is used to derive a new smoother algorithm based upon a decomposition of the linear model dynamical equation and maximizing use of rank 1 matrix modification this new algorithm it turns out parallels bierman s forward recursive square root information filter backward recursive u d factorized covariance algorithm the new result features computational efficiency reliance on numerically stable matrix modification algorithms and reduced computer storage

on the relationship between the lagrange multiplier method and the two filter smoother some relationships between the lagrange multiplier method and a two filter smoother are explored using orthogonal projection lemmas for linear discrele time systems the obtained expectation results are applied to algebraically derive a forward pass fixed interval smoothing algorithm that utilizes backward pass information filter outputs furthermore a convenient form for numerically efficient and stable algorithms is developed and the derived equation can also be used to quantitatively evaluate the effect of initial statistics on the smoothed estimate numerical results pertaining to a linearized version for the in track motion of a satellite travelling in a circular orbit are included to illustrate the use of the given algorithms

convergence of a nonconforming multiscale finite element method the multiscale finite element method msfem t y hou x h wu and z cai math comp 1998 to appear t y hou and x h wu j comput phys 134 1997 pp 169 189 has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients this is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator our previous study reveals that the leading order error in this approach is caused by the resonant sampling which leads to large error when the mesh size is close to the small scale of the continuous problem similar difficulty also arises in numerical upscaling methods an oversampling technique has been introduced to alleviate this difficulty t y hou and x h wu j comput phys 134 1997 pp 169 189 a consequence of the oversampling method is that the resulting finite element method is no longer conforming here we give a detailed analysis of the nonconforming error our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale we show that the cell resonance error is of lower order our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice

combining and updating of local estimates and regional maps along sets of one dimensional tracks in this paper we consider the problem of combining and updating estimates that may have been generated in a distributed fashion or may represent estimates generated at different times of the same process sample path the first of these cases has applications in decentralized estimation while the second has applications in updating maps of spatially distributed random quantities given measurements along several tracks the method of solution for the second problem uses the result of the first and the similarity in the formulation and solution of these problems emphasizes the conceptual similarity between many problems in decentralized control and in the analysis of random fields

proper orthogonal decomposition method for multiscale elliptic pdes with random coefficients abstract in this paper we develop an efficient multiscale reduced basis method to solve elliptic pdes with multiscale and random coefficients in a multi query setting our method consists of offline and online stages in the offline stage a small number of reduced multiscale basis functions are constructed within each coarse grid block using the proper orthogonal decomposition pod method moreover local tensor spaces are defined to approximate the solution space of the multiscale random pdes in the online stage a weak formulation is derived and discretized using the galerkin method to compute the solution since the multiscale reduced basis functions can efficiently approximate the high dimensional solution space our method is very efficient in solving multiscale elliptic pdes with random coefficients convergence analysis of the proposed method is presented which shows the dependence of the numerical error on the number of snapshots and the truncation threshold in the pod method finally numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale problems with or without scale separation in the physical space

harmonic oscillator pre potentials in su 2 lattice gauge theory we write the su 2 lattice gauge theory hamiltonian in d dimensions in terms of pre potentials which are su 2 fundamental doublets of harmonic oscillators the hamiltonian in terms of pre potentials has su 2 u 1 local gauge invariance these pre potentials enable us to solve the su 2 gauss law and characterize the su 2 gauge invariant hilbert space in terms of a set of integers we discuss the consequences of the additional u 1 gauge invariance the extension to su n lattice gauge theory is discussed

a heterogeneous stochastic fem framework for elliptic pdes we introduce a new concept of sparsity for the stochastic elliptic operator div a x which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure this new concept of sparsity motivates a heterogeneous stochastic finite element method hsfem framework for linear elliptic equations which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space we also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques the resulting hsfem involves two stages and suits the multi query setting in the offline stage the local stochastic structure of the solution space is identified in the online stage the equation can be efficiently solved for multiple forcing functions an online error estimation and correction procedure through monte carlo sampling is given numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the hsfem in the online stage

a note on factoring unitary matrices abstract the note concerns three factorizations of unitary matrices a recent factorization allowing to obtain any unitary matrix of dimension 2n from four unitary matrices of dimension n the cosine sine decomposition and a new factorization decomposing a given unitary matrix into a product of two orthogonal matrices and a diagonal unitary matrix

construction and applications of discrete time smoothing error models we present a unified perspective on techniques for constructing both forward and backward markovian realizations of the error process associated with discrete time fixed interval smoothing several alternative approaches are presented each of which provides additional insight and connections with other results in linear estimation theory in addition two applications of such models are described the first is a problem in the validation of an error model for a measurement system while the second concerns the problems of updating and combining smoothed estimates we also present an example of the use of our updating solution for the mapping of random fields based on several sets of data tracks

factorization of unitary matrices factorization of n n unitary matrices as a product of n diagonal phase matrices interlaced with n 1 orthogonal matrices each one generated by a real vector is provided as a byproduct an explicit form for the weyl factorization of unitary matrices is given the results can be used at the parametrization of complex hadamard matrices and in finding the laplace beltrami operators on unitary groups

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

a calculus for su 3 leading to an algebraic formula for the clebsch gordan coefficients we develop a simple computational tool for su 3 analogous to bargmann s calculus for su 2 crucial new inputs are i explicit representation of the gelfand zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable ii an auxiliary gaussian measure with respect to which the gelfand zetlin states are orthogonal but not normalized iii simple generating functions for generating all basis states and also all invariants as an illustration of our techniques an algebraic formula for the clebsch gordan coefficients is obtained for the first time this involves only gaussian integrations thus su 3 is made as accessible for computations as su 2 is

quasi monte carlo methods for elliptic pdes with random coefficients and applications we devise and implement quasi monte carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients our motivation comes from fluid flow in random porous media where relevant functionals include the fluid pressure velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field our emphasis is on situations where a very large number of random variables is needed to model the coefficient field as an alternative to classical monte carlo we here employ quasi monte carlo methods which use deterministically chosen sample points in an appropriate usually high dimensional parameter space each realization of the pde solution requires a finite element fe approximation in space and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid not necessarily the same as the fe grid in the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using fft in this way we avoid the use of a truncated karhunen loeve expansion but introduce high nominal dimension in parameter space numerical experiments with 2 dimensional rough random fields high variance and small length scale are reported showing that the quasi monte carlo method consistently outperforms the monte carlo method with a smaller error and a noticeably better than o n 1 2 convergence rate where n is the number of samples moreover the rate of convergence of the quasi monte carlo method does not appear to degrade as the nominal dimension increases examples with dimension as high as 10 6 are reported

coherent states and the classical limit on irreducible representations we give an explicit construction of the coherent states for an arbitrary irreducible representation we also construct the symplectic structure on the manifold of coherent states find canonical variables and discuss various classical limits of quantum mechanical systems with relevant observables that obey commutation relations

su n irreducible schwinger bosons we construct su n irreducible schwinger bosons satisfying certain u n 1 constraints which implement the symmetries of su n young tableaues as a result all su n irreducible representations are simple monomials of n 1 types of su n irreducible schwinger bosons further we show that these representations are free of multiplicity problems thus all su n representations are made as simple as su 2

brief square root information filtering and fixed interval smoothing with singularities the square root information filter algorithm and the related covariance smoother algorithm have been generalized to handle singular state transition matrices and perfect measurements this allows the use of srif techniques for problems with delays and state constraints the generalized algorithms use complete qr factorization to isolate deterministically known parts of the state and nonsingular parts of the state transition and disturbance influence matrices these factorizations and the corresponding changes of coordinates are used to solve the recursive least squares problems that are basic to the srif technique numerical stability computation time and storage requirements are comparable to the traditional srif algorithms

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

models for su 3 in terms of so n 2 and so 2n algebras mathematical and physical models for su 3 are given in terms of unirreps of the so n 2 and so 2n algebras vector coherent state representations are given for all the discrete series unirreps of so n 2 and so 2n traceless bosons are shown to arise naturally in these representations a new and elegant realisation of the so 8 approximately so 6 2 model for su 3 is given and it is shown how to perform calculations in the so 8 model using the powerful vector coherent state representations formalism

a su 3 wigner function for three dimensional systems we derive a wigner function with su 3 symmetry for three dimensional systems exemplified by the polarization of three dimensional electromagnetic fields

the canonical coset decomposition of unitary matrices through householder transformations this paper reveals the relation between the canonical coset decomposition of unitary matrices and the corresponding decomposition via householder reflections these results can be used to parametrize unitary matrices via householder reflections

coherent states for su 3 we define coherent states for su 3 using six bosonic creation and annihilation operators these coherent states are explicitly characterized by six complex numbers with constraints for the completely symmetric representations n 0 and 0 m only three of the bosonic operators are required for mixed representations n m all six operators are required the coherent states provide a resolution of identity satisfy the continuity property and possess a variety of group theoretic properties we introduce an explicit parametrization of the group su 3 and the corresponding integration measure finally we discuss the path integral formalism for a problem in which the hamiltonian is a function of su 3 operators at each site

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

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

su 3 revisited the d matrices for all states of the two fundamental representations and octet are given in the euler angle parametrization of su 3 the raising and lowering operators are given in terms of linear combinations of the left invariant vector fields of the group manifold in this parametrization using these differential operators the highest weight state of an arbitrary irreducible representation is found and a description of the calculation of clebsch gordon coefficients is given

wiener chaos expansions and numerical solutions of randomly forced equations of fluid mechanics in this paper we propose a numerical method based on wiener chaos expansion and apply it to solve the stochastic burgers and navier stokes equations driven by brownian motion the main advantage of the wiener chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner the separation principle effectively reduces a stochastic equation to its associated propagator a system of deterministic equations for the coefficients of the wiener chaos expansion simple formulas for statistical moments of the stochastic solution are presented these formulas only involve the solutions of the propagator we demonstrate that for short time solutions the numerical methods based on the wiener chaos expansion are more efficient and accurate than those based on the monte carlo simulations

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

su n coherent states we generalize schwinger boson representation of su 2 algebra to su n and define coherent states of su n using 2 2n 1 1 bosonic harmonic oscillator creation and annihilation operators we give an explicit construction of all n 1 casimirs of su n in terms of these creation and annihilation operators the su n coherent states belonging to any irreducible representations of su n are labeled by the eigenvalues of the casimir operators and are characterized by n 1 complex orthonormal vectors describing the su n manifold the coherent states provide a resolution of identity satisfy the continuity property and possess a variety of group theoretic properties

coherent states with su 2 and su 3 charges we define coherent states carrying su 2 charge by exploiting the schwinger boson representation of the su 2 lie algebra these coherent states satisfy the continuity property and provide resolution of identity on s3 we further generalize these techniques to construct the corresponding su 3 charge coherent states the su n extension is also discussed

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

quasi monte carlo finite element methods for elliptic pdes with lognormal random coefficients in this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in mathbb r d rd d 1 2 3 d 1 2 3 with diffusion coefficient a varvec x omega a x given as a lognormal random field i e a varvec x omega exp z varvec x omega a x exp z x where varvec x x is the spatial variable and z varvec x cdot z x is a gaussian random field the analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from 0 0 or infty over all possible realizations of a a focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem we give a rigorous error analysis for methods constructed from 1 standard continuous and piecewise linear finite element approximation in physical space 2 truncated karhunen loeve expansion for computing realizations of a a leading to a possibly high dimensional parametrized deterministic diffusion problem and 3 lattice based quasi monte carlo qmc quadrature rules for computing integrals over parameter space which define the expected values the paper contains novel error analysis which accounts for the effect of all three types of approximation the qmc analysis is based on a recent result on randomly shifted lattice rules for high dimensional integrals over the unbounded domain of euclidean space which shows that under suitable conditions the quadrature error decays with mathcal o n 1 delta o n 1 with respect to the number of quadrature points n n where delta 0 0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration

square root bryson frazier smoothing algorithms some new square root algorithms for bryson frazier smoothing formulas are suggested square root algorithms and a fast square root or so called chandrasekhar type algorithm the new square root algorithms use square root arrays composed of smoothed estimates and their error covariances these algorithms provide many advantages over the conventional algorithms with respect to systolic array and parallel implementations as well as numerical stability and conditioning for the case of constant parameter systems a fast square root algorithm is suggested which requires less computation than others

covariance factorization algorithms for fixed interval smoothing of linear discrete dynamic systems efficient factorized covariance smoothers designed to work with factorized covariance filters are derived for linear discrete dynamic systems the approach to factorized covariance smoothers either u d or square root uses outputs from factorized covariance filters and is closely derived from the g j bierman s earlier algorithm 1974 the dyer mcreynolds covariance smoother these algorithms are more efficient than the bierman s newer smoother 1983 based upon rank 1 process noise updates the efficiency of the new algorithms increases significantly as the order of process noise increases for full process noise they can be implemented in a way that avoids the inverse of the transition matrix

a new forward pass fixed interval smoother using the u d information matrix factorization abstract a new u d factorized smoothing algorithm that is numerically stable and reliable is developed by using a forward pass fixed interval smoother recursion introducing a gain for a backward pass information filter and the notion of an input to the smoother and decomposing the system noises into each element it is shown that three famous u d algorithms can be naturally applied to construct such a smoother the new result does not necessitate two burdensome matrix inversions or any algebraic computations equivalent to them for the transition matrix and the predicted error covariance which are at present necessary for the bierman s u d smoother consequently compared with the result of bierman the present algorithm can deal with a broader system which may cover a time delay system and provide an improvement in computation speed and computer storage