on the p and hp extension of nedelec s curl conforming elements abstract we establish interpolation error estimates for the hp extension of nedelec s curl and divergence conforming three dimensional finite elements using these interpolation estimates we investigate the use of higher order edge finite elements for the linear three dimensional magnetostatic problem our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p

the h text p version of the finite element method for domains with curved boundaries this paper analyzes the convergence of the h text p version of the finite element method for elliptic problems on curved domains it proves that when the boundary of the domain is piecewise analytic boundary conditions are piecewise analytic and the sides of the elements are analytic the h text p version has an exponential rate of convergence numerical computations are included the paper generalizes the results previously obtained by the authors the h text p version of finite element method part 1 the basic approximation results comp mech 1 1986 pp 21 41 the h text p version of finite element method part 2 general results and applications comp mech 1 1986 pp 203 220

recursive algorithms for identification of impulse noise channels the class a middleton model is a widely accepted statistical physical parameteric model for impulsive interference superimposed on a gaussian background in the present work a recursive decision directed estimator for online identification of the parameters of the class a model is proposed this estimator is based on an adaptive bayesian classification of each of a sequence of class a envelope samples as an impulsive sample or as a background sample as each sample is so classified recursive updates of the estimates of the second moment of the background component of the interference envelope density the second moment of the impulsive component of the interference envelope density and the probability with which the impulsive component occurs are readily obtained from these estimates estimates of the parameters of the class a model follow straightforwardly since closed form expressions for the parameters exist in terms of these quantities the performance characteristics of this algorithm are investigated and an appropriately modified version is found to yield a recursive algorithm with excellent global performance

the volume of gaussian states by information geometry we formulate the problem of determining the volume of the set of gaussian physical states in the framework of information geometry this is done by considering phase space probability distributions parametrized by their covariances and endowing the resulting statistical manifold with the fisher rao metric we then evaluate the volume of classical quantum and quantum entangled states for two mode systems showing chains of strict inclusions

a quasi separation principle and newton like scheme for coherent quantum lqg control abstract this paper is concerned with the coherent quantum linear quadratic gaussian lqg control problem of constructing an optimal controller for linear quantum plants using a quadratic performance criterion coherent quantum feedback control does not employ classical measurements which inherently entail the loss of quantum information a coherent quantum controller is itself a quantum system and this imposes physical realizability pr constraints on the quantum stochastic differential equation which governs such a controller pr corresponds to the equivalence of the controller to an open quantum harmonic oscillator whereby its state space matrices are related to the hamiltonian and coupling operators of the oscillator the hamiltonian parameterization of the controller is combined with frechet differentiation of the lqg cost with respect to the state space matrices in order to obtain equations for the optimal controller a quasi separation property for the gain matrices of the quantum controller is established and a newton like iterative scheme for the numerical solution of the equations is outlined

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

robust wiener filtering for multiple inputs with channel distortion corresp robust wiener filtering has previously been considered for the single input scalar case where there is no channel distortion and where the signal to be estimated is the source signal itself here these results are extended to the multiple input vector case where linear channel distortion is allowed and the signal to be estimated is a linear filtered version of the source signal the results are obtained from those for the single input ease by modifying the constraints on signal and noise characteristics such a modification is motivated by examining the expression of the mean squared error for the optimum filter

an hp error analysis of mitc plate elements we give an hp error analysis of several rectangular families of finite elements for the reissner mindlin plate bending equations we consider both the original mitc families k j bathe f brezzi and m fortin internat j numer methods engrg 28 1989 pp 1787 1801 and some new ones introduced in this paper for the deflection and rotation we give error estimates which are optimal with respect to the mesh size h and optimal up to ordo p varepsilon varepsilon 0 arbitrary with respect to the polynomial degree p we also obtain estimates for the error in the shear force calculated via two different methods our analysis utilizes some recent results of ours for the mixed method for the stokes problem as well as hp interpolation estimates for mixed methods for second order elliptic equations in this regard we derive new hp results in this paper for the brezzi douglas fortin marini spaces and improve upon previous estimates for the brezzi douglas marini spaces

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

quantum classical correspondence in perturbed chaotic systems we discuss the behavior of fidelity for a classically chaotic quantum system we show the existence of a critical value of the perturbation above which the quantum decay exponential or power law follows the classical one the independence of the decay rate of the perturbation strength discussed in the literature is a consequence of the quantum classical correspondence of the relaxation process

polynomial interpolation results in sobolev spaces abstract in this paper we derive new results related to the lagrange polynomial interpolation on the nodes of gauss type quadrature formulas in particular we obtain asymptotic error estimates of the same order as the distance to the best fit in the polynomial space

unsymmetric multi level hanging nodes and anisotropic polynomial degrees in h1 conforming higher order finite element methods abstract the implementation of higher order finite element schemes that can handle multi level hanging nodes is known to be a difficult task in fact most of the available literature on hanging nodes in finite element schemes restricts to one level hanging nodes resulting from symmetric bisections the intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a h 1 conforming higher order finite element method the corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric multi level hanging nodes and arbitrary anisotropic polynomial degree distributions where special care is given to possible orientation problems the meshes may even result from non matching refinements given these data structures and algorithms an extension for serendipity spaces is described in detail along with some other techniques to improve computational efficiency numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher order finite element schemes have to offer to the best of our knowledge this paper offers the most comprehensive numerical results so far for various three dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as multi level hanging nodes most notably the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates however the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen full tensor product or serendipity finite element spaces in addition to these numerical results we find that systems of equations defined via finite element spaces using the nodal lagrange polynomials with gauss lobatto quadrature points as support points yield drastically improved condition numbers

finite element methods principles for their selection abstract principles for the selection of a finite element method for a particular problem are discussed these principles are stated in terms of the notion of approximability optimality and stability several examples are discussed in detail as illustrations conclusions regarding the selection of finite element methods are summarized in the final section of the paper

separability dynamics of two mode gaussian states in parametric conversion and amplification we give a simplified form of simon s separability criterion for two mode gaussian states showing that for systems whose unitary evolution is governed by arbitrary time dependent quadratic hamiltonians the separability dynamics is completely described in terms of the determinant of the cross covariance matrix as concrete examples we consider the evolution of the inverse negativity coefficient which gives a quantitative estimation of the degree of entanglement for two initially uncoupled modes each being in a squeezed thermal state in the cases of parametric converter parametric amplifier and for a cavity whose boundary oscillates in resonance with two field modes

geometry of gaussian quantum states we study the hilbert schmidt measure on the manifold of mixed gaussian states in multi mode continuous variable quantum systems an analytical expression for the hilbert schmidt volume element is derived its corresponding probability measure can be used to study typical properties of gaussian states it turns out that although the manifold of gaussian states is unbounded an ensemble of gaussian states distributed according to this measure still has a normalizable distribution of symplectic eigenvalues from which unitarily invariant properties can be obtained by contrast we find that for an ensemble of one mode gaussian states based on the bures measure the corresponding distribution cannot be normalized as important applications we determine the distribution and the mean value of von neumann entropy and purity for the hilbert schmidt measure

cancellation of polarized impulsive noise using an azimuth dependent conditional mean estimator the separation of signals from noisy vector measurements is obtained by taking advantage of the middleton class a model of noise amplitude and the correlation of the components of the noise process due to their polarization the signal is assumed to be white gaussian noise is a superposition of m non gaussian processes each with a fixed azimuth of polarization neither the number of processes m nor their azimuths are known the separation of signal from noise is based on the conditional mean estimators in addition to the optimum estimator which can be derived from a knowledge of the bivariate density functions two suboptimum solutions for polarized noise are discussed the circularly symmetric estimator and the azimuth dependent one circular symmetry is suitable for the nonpolarized noise vector whereas the azimuth dependent estimator is tailored to polarized noise the azimuth dependent approach consists of two steps first the data vector process is discretized into azimuth sectors and then in those classified as noisy the signal is separated from the noise statistical model parameters of random processes are estimated by using the optimum classification based on the likelihood ratio test decision directed method iterative whitening methods are also discussed for correlated vector signals numerical examples show the effectiveness of the above technique in canceling polarized noise

negativity witness for the quantum ergodic conjecture for an ergodic system the time average of a classical observable coincides with that obtained via the liouville probability density a delta function on the energy shell reinterpreting this distribution as a wigner function that is the weyl representation of a density operator the quantum ergodic conjecture identifies this classical construction as an approximate representation of the quantum eigenstate of the same energy it is found that this reasonable hypothesis as far as expectations for observables are concerned does not satisfy the requirement of positivity so that the delta function on the shell cannot be a true wigner function this result was first presented by n balazs for the special case of a single degree of freedom such that the system is simultaneously integrable and ergodic here it is shown for a delta function on an arbitrary curved energy shell in a dimensional phase space that there exists a positive operator for which the conjectured wigner function predicts a negative expectation

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

entanglement criteria for all bipartite gaussian states we provide a necessary and sufficient condition for separability of gaussian states of bipartite systems of arbitrarily many modes the condition provides an operational criterion since it can be checked by simple computation moreover it allows us to find a pure product state decomposition of any given separable gaussian state we also show that all bipartite gaussian states with nonpositive partial transpose are distillable

partial scaling transform of multiqubit states as a criterion of separability the partial scaling transform of the density matrix for multiqubit states is introduced to detect entanglement of the quantum state the transform contains partial transposition as a special case the scaling transform corresponds to partial time scaling of a subsystem or partial planck constant scaling which was recently used to formulate a separability criterion for continuous variables a measure of entanglement which is a generalization of negativity measure is introduced being based on tomographic probability description of spin states

hp approximation theory for bdfm and rt finite elements on quadrilaterals we study approximation properties of hp finite element subspaces of boldsymbol mathsf h mathop rm div omega and boldsymbol mathsf h mathop rm rot omega on a polygonal domain omega using brezzi douglas fortin marini bdfm or raviart thomas rt elements approximation theoretic results are derived for the hp version finite element method on non quasi uniform meshes of quadrilateral elements with hanging nodes for functions belonging to weighted sobolev spaces boldsymbol mathsf h omega s ell omega and the countably normed spaces pmb cal b w ell omega these results culminate in a proof of the characteristic exponential convergence property of the hp version finite element method on suitably designed meshes under similar conditions needed for the analysis of the boldsymbol mathsf h 1 omega case by way of illustration exponential convergence rates are deduced for mixed hp approximation of flow in porous media

the optimal rate of convergence of the p version of the boundary element method in two dimensions we introduce the jacobi weighted besov and sobolev spaces in the one dimensional setting in the framework of these spaces we analyze lower and upper bounds for approximation errors in the p version of the boundary element method for hypersingular and weakly singular integral operators on polygons we prove the optimal rate of convergence for the p version in the energy norms of and respectively

a combined mixed and discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media the flow equation is solved by the mixed finite element method and the transport equation is approximated by a discontinuous galerkin method the scheme is continuous in time and a priori hp error estimates is presented

analysis for quadrilateral mitc elements for the reissner mindlin plate problem the present paper is made up of two parts in the first part we study the mathematical stability and convergence of the quadrilateral mitc elements for the reissner mindlin plate problem in an abstract setting we generalize the brezzi bathe fortin conditions to the quadrilateral mitc elements by weakening the second and fourth conditions under these conditions we show the well posedness of the discrete problem and establish an abstract error estimate in the energy norm the conclusion of this part is sparsity in the mathematical research of the quadrilateral mitc elements in the sense that one only needs to check these five conditions in the second part we extend four families of rectangular mitc elements of stenberg and suri to the quadrilateral meshes we prove that these quadrilateral elements satisfy the generalized brezzi bathe fortin conditions from the first part we develop the h p error estimates in both energy and l 2 norm for these quadrilateral elements for the first three families of quadrilateral elements the error estimates indicate that their convergent rates in both energy and l 2 norm depend on the mesh distortion parameter a we can get optimal error estimates for them provided that a 1 in addition we show the optimal convergence rates in energy norm uniformly in a for the fourth family of quadrilateral elements like their rectangular counterparts these quadrilateral elements are locking free

the h p and h p versions of the finite element methods in 1 dimension part iii the adaptive h p version abstract the paper is the third and final part in the series of three devoted to the detailed analysis of the three basic versions of the finite element method in one dimension the first part analyzed the p version and the second part concentrated on the h and h p version this paper analyzes a theoretical frame of the adaptive h p version and based on it the authors provide concrete algorithm for the one dimensional problem it is proven that in the case that the solution has x sub alpha type singularity the adaptive algorithm give an exponential rate of convergence very close to the optimal one analyzed in the second part of the paper additional keyword error analysis

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

parameter estimation for middleton class a interference processes the problem of estimating the parameters of the middleton class a interference model is considered on the assumption of the availability of a set of independent samples from class a envelope distribution the asymptotic performances of several estimation procedures are explored from this analysis estimates based on the method of moments are seen to be consistent and computationally desirable but highly inefficient whereas more efficient likelihood based estimators are seen to be computationally unwieldy however an estimator that initiates likelihood iteration with the method of moments estimates is seen to overcome these difficulties in its asymptotic performance unfortunately simulation of this third estimator for practical sample sizes reveals poor performance under these conditions to overcome this lack of small sample efficiency a similar estimator that initiates likelihood iteration with physically motivated but nonoptimal estimates is also proposed simulation of this latter estimator for practical sample sizes indicates that near optimal performance is attained by this technique

initial or final values for semiclassical evolutions in the weyl wigner representation initial value representations are constructed to avoid the search for trajectories that are only defined in semiclassical approximations by their boundary conditions we show how to incorporate these procedures within the full weyl representation so that quantum expectation values are given by phase space integrals over the evolving wigner function spurious semiclassical singularities at caustics are cancelled even though there is no increase in the number of trajectories as compared to usual semiclassical formulae the whole construction remains exact in the case of quadratic hamiltonians the evolution of density operators depends on either a forward and backward trajectory given by an initial value or else on a pair of trajectories propagating backwards from their final value the latter option reduces numerical errors in the computation of trajectories the general scheme also leads to analogous algorithms for evolving the quantum fidelity which can be approximated perturbatively with a single trajectory reducing to the dephasing representation for small times the theory is developed within a generalized maslov method based on semiclassical fourier transforms in order to avoid singularities in the limit of small times

toward a universal h p adaptive finite element strategy part 2 a posteriori error estimation abstract in this second paper of the series on h p adaptive finite element methods we present five different techniques for a posteriori error estimation of h p methods for linear elliptic boundary value problems the techniques are referred to as the residual estimation method the duality method the subdomain residual method a method based on interpolation theory and a post processing method the present study examines the theory and behavior of such techniques applied specifically to h p adaptive finite element methods several numerical examples illustrate the efficiency of the methods

the approximation theory for the p version finite element method and application to non linear elliptic pdes approximation theoretic results are obtained for approximation using continuous piecewise polynomials of degree p on meshes of triangular and quadrilateral elements estimates for the rate of convergence in sobolev spaces w m q omega q in 1 infty are given the results are applied to estimate the rate of convergence when the p version finite element method is used to approximate the alpha laplacian it is shown that the rate of convergence of the p version is always at least that of the h version measured in terms of number of degrees of freedom used if the solution is very smooth then the p version attains an exponential rate of convergence if the solution has certain types of singularity the rate of convergence of the p version is twice that of the h version the analysis generalises the work of babuska and others to the case q not 2 in addition the approximation theoretic results find immediate application for some types of spectral and spectral element methods

adaptive energy minimisation for h p finite element methods this article is concerned with the numerical solution of convex variational problems more precisely we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in an adaptive manner specifically we outline a new approach in the context of h p adaptive finite element methods employed for the efficient numerical solution of linear and nonlinear second order boundary value problems numerical experiments are presented which highlight the practical performance of this new h p refinement technique for both one and two dimensional problems

exchangeable stationary and entangled chains of gaussian states we explore conditions on the covariance matrices of a consistent chain of mean zero finite mode gaussian states in order that the chain may be exchangeable or stationary for an exchangeable chain our conditions are necessary and sufficient every stationary gaussian chain admits an asymptotic entropy rate whereas an exchangeable chain admits a simple expression for its entropy rate in our examples of stationary chains the same admits an integral formula based on the asymptotic eigenvalue distribution for toeplitz matrices an example of a stationary entangled gaussian chain is given

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

entanglement detection complexity and shannon entropic criteria entanglement plays a key role in quantum information the separability problem arises how to prove that a given quantum state is entangled covariance matrix based entanglement criteria are the most important and widely used approach to this problem in continuous variable systems here the separability problem is systematically studied in the more experimentally realistic setting that the questioned state is only partly known a simple general formalism is proposed for this variant of the separability problem it builds relations between certain classes of criteria and bounds the power of these criteria in particular it shows the equivalence of many seemingly different covariance matrix based criteria it offers guidance on the study of and reveals the inherent limitations of covariance matrix based criteria also a shannon entropic criterion is proposed for m n mode states it unifies and improves many known covariance matrix based criteria and shannon entropic criteria it is necessary and sufficient for 1 n mode gaussian states and m n mode bisymmetric gaussian states

a criterion for local interconvertibility between all tripartite gaussian states we arrive at a necessary and sufficient criterion that can be readily used for interconvertibility between general all tripartite gaussian states under local quantum operation the derivation involves a systematic reduction that converts the original complex conditions in high dimensional 6n 6n matrix space eventually into 2 2 matrix problems

entropy power uncertainty relations towards a tight inequality for all gaussian pure states we show that a proper expression of the uncertainty relation for a pair of canonically conjugate continuous variables relies on entropy power a standard notion in shannon information theory for real valued signals the resulting entropy power uncertainty relation is equivalent to the entropic formulation of the uncertainty relation due to bialynicki birula and mycielski but can be further extended to rotated variables hence based on a reasonable assumption we give a partial proof of a tighter form of the entropy power uncertainty relation taking correlations into account and provide extensive numerical evidence of its validity interestingly it implies the generalized rotation invariant schrodinger robertson uncertainty relation exactly as the original entropy power uncertainty relation implies heisenberg relation it is saturated for all gaussian pure states in contrast with hitherto known entropic formulations of the uncertainty principle

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

semiclassical theory for small displacements characteristic functions contain complete information about all the moments of a classical distribution and the same holds for the fourier transform of the wigner function a quantum characteristic function or the chord function however knowledge of a finite number of moments does not allow for accurate determination of the chord function this provides the overlap of the state with all its possible rigid translations or displacements we here present a semiclassical approximation of the chord function for large bohr quantized states which is accurate right up to a caustic beyond which the chord function becomes evanescent it is verified to pick out blind spots which are displacements for zero overlaps these occur even for translations within a planck area of the origin we derive a simple approximation for the closest blind spots depending on the schrodinger covariance matrix which is verified for bohr quantized states

pure state correlations chords in phase space the intensity of the overlap of a quantum state with all its phase space translations defines its quantum correlations in the case of pure states these are invariant with respect to fourier transformation the overlaps themselves are here studied in terms of the wigner function and its fourier transform i e the characteristic function or chord function unlike the wigner function the chord function need not be real but eventual symmetry with respect to reflections about a phase space point may relate these representations semiclassical approximations for the classical like region of small chords and for large chords are derived these lead to an interpretation of the fourier invariance in terms of conjugate chords the interrelation of large and small sub planck phase space structures previously noted in the literature is thus reinterpreted

synthesis of linear coherent quantum control systems using a differential evolution algorithm we propose a new method to construct an optimal linear coherent quantum controller based on an evolutionary optimization method namely a differential evolution algorithm the aim is to provide a straightforward approach to deal with both nonlinear and nonconvex constraints arising in the coherent quantum controller synthesis the solution to this control problem involves a complex algebraic riccati equation which corresponds to a physical realizability condition for the coherent quantum controller the proposed method is demonstrated through an example of an entanglement control problem for a quantum network comprising two cascaded optical parametric amplifiers

peres horodecki separability criterion for continuous variable systems the peres horodecki criterion of positivity under partial transpose is studied in the context of separability of bipartite continuous variable states the partial transpose operation admits in the continuous case a geometric interpretation as mirror reflection in phase space this recognition leads to uncertainty principles stronger than the traditional ones to be obeyed by all separable states for all bipartite gaussian states the peres horodecki criterion turns out to be a necessary and sufficient condition for separability

hellinger distance as a measure of gaussian discord especially investigated in recent years the gaussian discord can tentatively be quantified by a distance between a given two mode gaussian state gs and the set of all the zero discord two mode gss however as this set consists only of product states such a distance captures all the correlations quantum and classical between modes therefore it is merely an upper bound for the geometric discord no matter which is the employed distance in this work we choose for this purpose the hellinger metric that is known to have many beneficial properties recommending it as a good measure of quantum behaviour in general this metric is determined by affinity a relative of the uhlmann fidelity with which it shares many important features as a first step of our work the affinity of a pair of n mode gss is written then in the two mode case we succeeded in determining exactly the closest gaussian product state and computed the gaussian discord accordingly the obtained general formula is remarkably simple and becomes still friendlier in the significant case of symmetric two mode gss we then analyse in detail two special classes of two mode gss of theoretical and experimental interest as well the squeezed thermal states and the mode mixed thermal ones the former are separable under a well known threshold of squeezing while the latter are always separable it is worth stressing that for symmetric states belonging to either of these classes we find consistency between their geometric hellinger discord and the originally defined discord in the gaussian approach at the same time the gaussian hellinger discord of such a state turns out to be a reliable measure of the total amount of its cross correlations

the hp mitc finite element method for the reissner mindlin plate problem the popular mitc finite elements used for the approximation of the reissner mindlin plate are extended to the case where elements of non uniform degree p distribution are used on locally refined meshes such an extension is of particular interest to the hp version and hp adaptive finite element methods a priori error bounds are provided showing that the method is locking free the analysis is based on new approximation theoretic results for non uniform brezzi douglas fortin marini spaces and extends the results obtained in the case of uniform order approximation on globally quasi uniform meshes presented by stenberg and suri siam j numer anal 34 1997 544 numerical examples illustrating the theoretical results and comparing the performance with alternative standard galerkin approaches are presented for two new benchmark problems with known analytic solution including the case where the shear stress exhibits a boundary layer the new method is observed to be locking free and able to provide exponential rates of convergence even in the presence of boundary layers

recursive filtering with non gaussian noises the kalman filter is an optimal recursive filter although its optimality can only be claimed under the gaussian noise environment in this paper we consider the problem of recursive filtering with non gaussian noises one of the most promising schemes which was proposed by masreliez 1972 1975 uses the nonlinear score function as the correction term in the state estimate unfortunately the score function cannot be easily implemented except for simple cases in this paper a new method for efficient evaluation of the score function is developed the method employs an adaptive normal expansion to expand the score function followed by truncation of the higher order terms consequently the score function can be approximated by a few central moments the normal expansion is made adaptive by using the concept of conjugate recentering and the saddle point method it is shown that the approximation is satisfactory and the method is simple and practically feasible experimental results are reported to demonstrate the effectiveness of the new algorithm

wigner functions on non standard symplectic vector spaces we consider the weyl quantization on a flat non standard symplectic vector space we focus mainly on the properties of the wigner functions defined therein in particular we show that the sets of wigner functions on distinct symplectic spaces are different but have non empty intersections this extends previous results to arbitrary dimension and arbitrary constant symplectic structure as a by product we introduce and prove several concepts and results on non standard symplectic spaces which generalize those on the standard symplectic space namely the symplectic spectrum williamson s theorem and narcowich wigner spectra we also show how wigner functions on non standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation

quantification of gaussian quantum steering einstein podolsky rosen steering incarnates a useful nonclassical correlation which sits between entanglement and bell nonlocality while a number of qualitative steering criteria exist very little has been achieved for what concerns quantifying steerability we introduce a computable measure of steering for arbitrary bipartite gaussian states of continuous variable systems for two mode gaussian states the measure reduces to a form of coherent information which is proven never to exceed entanglement and to reduce to it on pure states we provide an operational connection between our measure and the key rate in one sided device independent quantum key distribution we further prove that peres conjecture holds in its stronger form within the fully gaussian regime namely steering bound entangled gaussian states by gaussian measurements is impossible

symplectically covariant schrodinger equation in phase space a classical theorem of stone and von neumann states that the schrodinger representation is up to unitary equivalences the only irreducible representation of the heisenberg group on the hilbert space of square integrable functions on configuration space using the wigner moyal transform we construct an irreducible representation of the heisenberg group on a certain hilbert space of square integrable functions defined on phase space this allows us to extend the usual weyl calculus into a phase space calculus and leads us to a quantum mechanics in phase space equivalent to standard quantum mechanics we also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space schrodinger equation

on computing the pressure by thep version of the finite element method for stokes problem this paper introduces and analyzes two ways of extracting the hydrostatic pressure when solving stokes problem using thep version of the finite element method when one uses a localh 1 projection we show that optimal rates of convergence for the pressure approximation is achieved when the pressure is not inh 1 or the value of the pressure is only needed at a few points one may extract the pressure pointwise using e g a single layer potential recovery negative zero and higher norm estimates for the stokes velocity are derived within the framework of thep version of the f e m

dynamics of bipartite and tripartite entanglement in a dissipative system of continuous variables abstract with the help of the characteristic function and the covariance matrix we study the effect of a dissipative bath on bipartite entanglement in a system of continuous variables with no direct inter mode coupling under the dissipation induced by an agarwal bath the time evolution of entanglement between two oscillators initially in a squeezed state has been examined and an analytical criterion as a function of the squeezing parameter s and the bath temperature n has been derived for bipartite entanglement at long times resulting in an entanglement phase diagram comprising of a sudden death region and a no sudden death region if s n lies in the no sudden death region there will be sustained bipartite entanglement in the long time limit the analysis has been extended to three mode systems including fully symmetric tripartite states and bisymmetric states which are invariant with the exchange of two modes out of the three if two symmetric modes are propagated in separate baths the bipartite entanglement between one of the two and the third mode will vanish in a finite time the disappearance of the entanglement leads to a decrease in cloning fidelities to below the classical value of 0 5 for fully symmetric tripartite states both the separate and common bath cases have been considered it is found that entanglement vanishes in a finite time in the presence of separate baths while it persists for a long time in the presence of a common bath

einstein podolsky rosen like separability indicators for two mode gaussian states we investigate the separability of the two mode gaussian states tmgss by using the variances of a pair of einstein podolsky rosen epr like observables our starting point is inspired by the general necessary condition of separability introduced by duan et al 2000 phys rev lett 84 2722 we evaluate the minima of the normalized forms of both the product and sum of such variances as well as that of a regularized sum making use of simon s separability criterion which is based on the condition of positivity of the partial transpose ppt of the density matrix simon 2000 phys rev lett 84 2726 we prove that these minima are separability indicators in their own right they appear to quantify the greatest amount of epr like correlations that can be created in a tmgs by means of local operations furthermore we reconsider the epr like approach to the separability of tmgss which was developed by duan et al with no reference to the ppt condition by optimizing the regularized form of their epr like uncertainty sum we derive a separability indicator for any tmgs we prove that the corresponding epr like condition of separability is manifestly equivalent to simon s ppt one the consistency of these two distinct approaches epr like and ppt affords a better understanding of the examined separability problem whose explicit solution found long ago by simon covers all situations of interest

mixed finite element methods for nonlinear elliptic problems the h p version mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed existence and uniqueness of the approximate solution are demonstrated using a fixed point argument convergence and stability of the method are proved both with respect to mesh refinement and increase of the degree of the approximating polynomials the analysis is carried out in detail using raviart thomas nedelec spaces as an example numerical results for minimal surface problems are obtained using brezzi douglas marini spaces graphs of the approximate solutions are presented for two different problems

adaptive scheme for elimination of broadband noise and impulsive disturbances from ar and arma signals the problem of elimination of measurement noise and impulsive disturbances from autoregressive and autoregressive moving average signals is considered it is shown that the task of simultaneous detection tracking restoration can be stated as a nonlinear filtering problem and solved using the theory of extended kalman filter numerical tests carried out for real audio signals corrupted by both real and artificially generated disturbances confirm very good properties of the proposed algorithm

a priori error analysis of stochastic galerkin mixed approximations of elliptic pdes with random data we construct stochastic galerkin approximations to the solution of a first order system of pdes with random coefficients under the standard finite dimensional noise assumption we transform the variational saddle point problem to a parametric deterministic one approximations are constructed by combining mixed finite elements on the computational domain with m variate tensor product polynomials we study the inf sup stability and well posedness of the continuous and finite dimensional problems the regularity of solutions with respect to the m parameters describing the random coefficients and establish a priori error estimates for stochastic galerkin finite element approximations

parity measurements decoherence and spiky wigner functions notwithstanding radical conceptual differences between classical and quantum mechanics it is usually assumed that physical measurements concern observables common to both theories not so with the eigenvalues 1 of the parity operator the effect of such a measurement on a mixture of even and odd states of the harmonic oscillator is akin to separating at a single stroke a pair of shuffled card decks the result is a set of definite parity though otherwise mixed here we derive the general form of a parity collapsed state whether pure or mixed the signature of positive or negative parity is a corresponding spike in the wigner function which is sharpened by decoherence we conjecture that states with pure parity always have negative values in their wigner functions

symplectic evolution of wigner functions in markovian open systems the wigner function is known to evolve classically under the exclusive action of a quadratic hamiltonian if the system also interacts with the environment through lindblad operators that are complex linear functions of position and momentum then the general evolution is the convolution of a non hamiltonian classical propagation of the wigner function with a phase space gaussian that broadens in time we analyze the consequences of this in the three generic cases of elliptic hyperbolic and parabolic hamiltonians the wigner function always becomes positive in a definite time which does not depend on the initial pure state we observe the influence of classical dynamics and dissipation upon this threshold we also derive an exact formula for the evolving linear entropy as the average of a narrowing gaussian taken over a probability distribution that depends only on the initial state this leads to a long time asymptotic formula for the growth of linear entropy we finally discuss the possibility of recovering the initial state

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

the p version of the finite element method for constraint boundary conditions abstract this paper addresses the implementation of general constraint boundary conditions for a system of equations by the p version of the finite element method by constraint boundary conditions the authors mean conditions where some relation between the components is prescribed at the boundary optimal error bounds are proven keywords convergence sobolev spaces

order reduction and how to avoid it when explicit runge kutta nystrom methods are used to solve linear partial differential equations in this paper we study the order reduction which turns up when explicit runge kutta nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines the order observed in practice including its fractional part is obtained it is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization the numerical experiments confirm that the optimal order can be reached

discrete wigner distribution for two qubits a characterization of entanglement properties we study the properties of the discrete wigner distribution for two qubits introduced by wotters in particular we analyse the entanglement properties within the wigner distribution picture by considering the negativity of the wigner function wf and the correlations of the marginal distribution we show that a state is entangled if at least one among the values assumed by the corresponding discrete wf is smaller than a certain critical negative value then based on the partial transposition criterion we establish a relation between the separability of a density matrix and the non negativity of the wfs relevant both to such a density matrix and to the partially transposed thereof finally we derive a simple inequality involving the covariance matrix elements of a given wf which appears to provide a separability criterion stronger than the one based on the local uncertainty relations

physical properties of the schur complement of local covariance matrices general properties of global covariance matrices representing bipartite gaussian states can be decomposed into properties of local covariance matrices and their schur complements we demonstrate that given a bipartite gaussian state 12 described by a 4 4 covariance matrix v the schur complement of a local covariance submatrix v1 of it can be interpreted as a new covariance matrix representing a gaussian operator of party 1 conditioned to local parity measurements on party 2 the connection with a partial parity measurement over a bipartite quantum state and the determination of the reduced wigner function is given and an operational process of parity measurement is developed generalization of this procedure to an n partite gaussian state is given and it is demonstrated that the n 1 system state conditioned to a partial parity projection is given by a covariance matrix such that its 2 2 block elements are schur complements of special local matrices

on structural physical approximations and entanglement breaking maps very recently a conjecture saying that the so called structural physical approximations spas to optimal positive maps optimal entanglement witnesses give entanglement breaking eb maps separable states has been posed korbicz et al 2008 phys rev a 78 062105 the main purpose of this contribution is to explore this subject first we extend the set of entanglement witnesses supporting the conjecture then we ask whether spas constructed from other than the depolarizing channel maps also lead to eb maps and show that in general this is not the case on the other hand we prove an interesting fact that for any positive map there exists an eb channel such that the spa of constructed with the aid of is again an eb channel finally we ask similar questions in the case of continuous variable systems we provide a simple way of constructing spa and prove that in the case of the transposition map it gives the eb channel

on the convergence rate of the cell discretization algorithm for solving elliptic problems error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both h k and analytic solutions to selfadjoint elliptic problems the polynomial implementation of this algorithm can be viewed as a nonconforming version of the h p finite element method that also can produce the continuous approximations of the h p method the examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results

moments of non gaussian wigner distributions and a generalized uncertainty principle i the single mode case the non negativity of the density operator of a state is faithfully coded in its wigner distribution and this coding places on the moments of the wigner distribution constraints arising from the non negativity of the density operator working in a monomial basis for the algebra of operators on the hilbert space of a bosonic mode we formulate these constraints in a canonically covariant form which is both concise and explicit since the conventional uncertainty relation is such a constraint on the first and second moments our result constitutes a generalization of the same to all orders the structure constants of in the monomial basis are shown to be essentially the su 2 clebsch gordan coefficients our results have applications in quantum state reconstruction using optical homodyne tomography and when generalized to the n mode case which will be done in the second part of this work will have applications also for continuous variable quantum information systems involving non gaussian states

inseparability of mixed two mode gaussian states generated with a su 1 1 interferometer we study the entanglement properties of two mode gaussian light emerging from a generic su 1 1 interferometer our tool is the two mode characteristic function which is determined by the 4 4 covariance matrix for an initial product of two mixed single mode gaussian states we investigate the output two mode covariance matrix its structure displays the noise properties of the reduced states as well as the correlations between modes classicality of the output two mode state is characterized by the existence of the glauber sudarshan p representation for testing separability we apply the peres simon criterion requiring preservation of the positivity of the density matrix under partial transposition since inseparability entails nonclassicality the threshold gain above which nonclassicality of the output state becomes manifest is lower than that allowing for its inseparability we find that only for a thermal input do nonclassicality and inseparability of the output have the same threshold gain

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

on the convergence of the p version of the boundary element galerkin method abstract the authors consider various physical problems which may be formulated in terms of integral equations of the first kind including the two dimensional screen neumann and dirichlet problems in acoustics and crack problems in elasticity sharp regularity results for the solutions are available for these problems proven is the convergence of the p version for some galerkin boundary element schemes based on the integral equation formulations it is shown that the rate of convergence obtained by our method is twice that for the usual h version

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

monogamy inequality for distributed gaussian entanglement we show that for all n mode gaussian states of continuous variable systems the entanglement shared among n parties exhibits the fundamental monogamy property the monogamy inequality is proven by introducing the gaussian tangle an entanglement monotone under gaussian local operations and classical communication which is defined in terms of the squared negativity in complete analogy with the case of n qubit systems our results elucidate the structure of quantum correlations in many body harmonic lattice systems

strong subadditivity for log determinant of covariance matrices and its applications we prove that the log determinant of the covariance matrix obeys the strong subadditivity inequality for arbitrary tripartite states of multimode continuous variable quantum systems this establishes general limitations on the distribution of information encoded in the second moments of canonically conjugate operators the inequality is shown to be stronger than the conventional strong subadditivity inequality for von neumann entropy in a class of pure tripartite gaussian states we finally show that such an inequality implies a strict monogamy type constraint for joint einstein podolsky rosen steerability of single modes by gaussian measurements performed on multiple groups of modes

on the stability and convergence of higher order mixed finite element methods for second order elliptic problems we investigate the use of higher order mixed methods for second order elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p our estimates yield asymptotic convergence rates for the p and h p versions of the finite element method they also describe more accurately than previously proved estimates the increased rate of convergence expected when the h version is used with higher order polynomials

entangling oscillators through environment noise we consider two independent bosonic oscillators immersed in a common bath evolving in time with a completely positive markovian quasi free gaussian reduced dynamics we show that an initially separated gaussian state can become entangled as a result of a purely noisy mechanism in certain cases the dissipative dynamics allows the persistence of these bath induced quantum correlations even in the asymptotic equilibrium state

optimal error estimation for bfh rmcurl conforming p interpolation in two dimensions in this paper we prove an optimal error estimate for the bf h rm curl conforming projection based p interpolation operator introduced in l demkowicz and i babuska siam j numer anal 41 2003 pp 1195 1208 this result is proved on the reference element either triangle or square k for regular vector fields in bf h r rm curl k with arbitrary r 0 the formulation of the result in the bf h rm div conforming setting which is relevant for the analysis of high order boundary element approximations for maxwell s equations is provided as well