a stream virtual element formulation of the stokes problem on polygonal meshes in this paper we propose and analyze a novel stream formulation of the virtual element method vem for the solution of the stokes problem the new formulation hinges upon the introduction of a suitable stream function space characterizing the divergence free subspace of discrete velocities and it is equivalent to the velocity pressure inf sup stable mimetic scheme presented in l beirao da veiga et al j comput phys 228 2009 pp 7215 7232 up to a suitable reformulation into the vem framework both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one

the virtual element method for discrete fracture network simulations abstract in this work an optimization based approach presented in berrone et al 2013 2014 10 12 for discrete fracture network simulations is coupled with the virtual element method vem for the space discretization of the underlying darcy law the great flexibility of the vem in handling rather general polygonal elements allows in a natural way for an effective description of irregular solutions starting from an arbitrary triangulation which is built independently of the mesh on other fractures only partial conformity is in fact obtained with this approach numerical results performed on several dfn configurations confirm the viability and efficiency of the resulting method

topology optimization using polytopes abstract meshing complex engineering domains is a challenging task arbitrary polyhedral meshes can provide the much needed flexibility in automated discretization of such domains the geometric property of polyhedral meshes such as its unstructured nature and the connectivity of faces between elements makes them specially attractive for topology optimization applications numerical anomalies in designs such as the single node connections and checkerboard pattern can be naturally circumvented with polyhedrons in the current work we solve the governing three dimensional elasticity state equation using the virtual element method vem approach the main characteristic difference between vem and standard finite element methods fem is that in vem the canonical basis functions are not constructed explicitly rather the stiffness matrix is computed directly utilizing a projection map which extracts the linear component of the deformation such a construction guarantees the satisfaction of the patch test used by engineers as an indicator of optimal convergence of numerical solutions under mesh refinement finally the computations reduce to the evaluation of matrices which contain purely geometric surface facet quantities the present work focuses on the first order vem in which the degrees of freedom are associated with the vertices the features of the current optimization approach are demonstrated using numerical examples for compliance minimization and compliant mechanism problems

topology optimization of incompressible media using mixed finite elements abstract we present an alternative formulation for the topology optimization of structures made of incompressible materials a topic that cannot be tackled using most of the approaches of the current literature that are mainly based on displacement finite elements that are well known to be affected by the locking phenomenon a way out of the problem has recently been proposed in sigmund and clausen 2007 o sigmund p m clausen topology optimization using a mixed formulation an alternative way to solve pressure load problems comput methods appl mech engrg 196 13 16 2007 1874 1889 based on a displacement pressure finite element discretization the approach presented here consists in a truly mixed variational formulation coupled to a mixed fem discretization that uses the composite element of johnson and mercier for the discretization of the stress field by so doing the continuous and discrete inf sup conditions of the problem are automatically met even in the presence of incompressible materials a few numerical studies are presented to validate the theoretical framework for which the well known method of moving asymptotes mma is adopted for the numerical optimization of the problem different topologies in plane stress and plane strain conditions are presented with particular attention to the convergence of the last ones to pure 0 1 designs some forthcoming investigations are eventually highlighted including the solution of stress constrained topology optimization problems that find in the truly mixed setting their natural environment and the extension of the present formulation to deal with pressure load structural problems

geometrical description of quantal state determination under the assumption that every quantal measurement may give data about the post measurement state of the inspected ensemble the problem of the state determination is reconsidered it is shown that orthogonal decomposition of the set of complex n n hermitian matrices into the commutative subsets allows operators to be found such that post measurement information on these observables allows a partial in some cases total determination of the pre measurement state to be effected

bipartite and tripartite entanglement of truncated harmonic oscillator coherent states via beam splitters we introduce a special class of truncated weyl heisenberg algebra and discuss the corresponding hilbertian and analytical representations subsequently we study the effect of a quantum network of beam splitting on coherent states of this nonlinear class of harmonic oscillators we particularly focus on quantum networks involving one and two beam splitters and examine the degree of bipartite as well as tripartite entanglement using the linear entropy

virtual element methods for hyperbolic problems on polygonal meshes in the present paper we develop the virtual element method for hyperbolic problems on polygonal meshes considering the linear wave equations as our model problem after presenting the semi discrete scheme we derive the convergence estimates in h 1 semi norm and l 2 norm moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the newmark method and the bathe method finally we show the practical behaviour of the proposed method through a large set of numerical tests

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

virtual element methods for plate bending problems abstract we discuss the application of virtual elements to linear plate bending problems in the kirchhoff love formulation as we shall see in the virtual element environment the treatment of the c 1 continuity condition is much easier than for traditional finite elements the main difference consists in the fact that traditional finite elements for every element k and for every given set of degrees of freedom require the use of a space of polynomials or piecewise polynomials for composite elements for which the given set of degrees of freedom is unisolvent for virtual elements instead we only need unisolvence for a space of smooth functions that contains a subset made of polynomials whose degree determines the accuracy as we shall see the non polynomial part of our local spaces does not need to be known in detail and therefore the construction of the local stiffness matrix is simple and can be done for much more general geometries

on the virtual element method for topology optimization on polygonal meshes it is well known that the solution of topology optimization problems may be affected both by the geometric properties of the computational mesh which can steer the minimization process towards local and non physical minima and by the accuracy of the method employed to discretize the underlying differential problem which may not be able to correctly capture the physics of the problem in light of the above remarks in this paper we consider polygonal meshes and employ the virtual element method vem to solve two classes of paradigmatic topology optimization problems one governed by nearly incompressible and compressible linear elasticity and the other by stokes equations several numerical results show the virtues of our polygonal vem based approach with respect to more standard methods

refined factorizations of solvable potentials a generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in quantum mechanics the method is applied in the study of radial oscillator morse and coulomb potentials to obtain a wide set of raising and lowering operators and to show clearly the connection that links these systems

factorization method and new potentials with the oscillator spectrum a one parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator this is a new derivation of a class of potentials previously obtained by abraham and moses with the help of the gelfand levitan formalism

alphabetic codes revisited an alphabetic code for an ordered probability distribution p sub k is a prefix code in which p sub k is assigned to the kth codeword of the coding tree in left to right order this class of codes is applied to binary test problems several earlier results on alphabetic codes are unified and enhanced the characteristic inequality for alphabetic codes that is analogous to the kraft inequality for prefix codes is also derived it is shown that if p sub k is in ascending or descending order l sub min the expected length of an optimal alphabetic code is the same as that of a huffman code for the unordered distribution p sub k an enhancement of gilbert and moore s 1959 merging property of all optimal alphabetic code is proved two lower bounds and a new upper bound on the expected length of an optimal alphabetic code are also proven and a simple method is proposed for constructing good alphabetic codes when optimality is critical

on the virtual element method for three dimensional linear elasticity problems on arbitrary polyhedral meshes abstract we explore the recently proposed virtual element method vem for the numerical solution of boundary value problems on arbitrary polyhedral meshes more specifically we focus on the linear elasticity equations in three dimensions and elaborate upon the key concepts underlying the first order vem while the point of departure is a conforming galerkin framework the distinguishing feature of vem is that it does not require an explicit computation of the trial and test spaces thereby circumventing a barrier to standard finite element discretizations on arbitrary grids at the heart of the method is a particular kinematic decomposition of element deformation states which in turn leads to a corresponding decomposition of strain energy by capturing the energy of linear deformations exactly one can guarantee satisfaction of the patch test and optimal convergence of numerical solutions the decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation as we show computing these projection maps and subsequently the local stiffness matrices in practice reduces to the computation of purely geometric quantities in addition to discussing aspects of implementation of the method we present several numerical studies in order to verify convergence of the vem and evaluate its performance for various types of meshes

filters in topology optimization in this article a modified filtered version of the minimum compliance topology optimization problem is studied the direct dependence of the material properties on its pointwise density is replaced by a regularization of the density field by the mean of a convolution operator in this setting it is possible to establish the existence of solutions moreover convergence of an approximation by means of finite elements can be obtained this is illustrated through some numerical experiments the filtering technique is also shown to cope with two important numerical problems in topology optimization checkerboards and mesh dependent designs copyright 2001 john wiley sons ltd

virtual element method for fourth order problems l 2 estimates abstract we analyse the family of c 1 virtual elements introduced in brezzi and marini 2013 for fourth order problems and prove optimal estimates in l 2 and in h 1 via classical duality arguments

coherent states for anharmonic oscillator hamiltonians with equidistant and quasi equidistant spectra two kinds of transformation for the time dependent schrodinger equation i e the differential and integral transformations are introduced if one considers only stationary solutions of this equation both transformations reduce to the well known darboux transformation for the stationary schrodinger equation when applied to non stationary solutions they give different results both transformations are invertible in appropriate spaces with the help of these transformations alternative systems of coherent states to those in the literature are obtained for isospectral hamiltonians with equidistant spectra these transformations are also applied to the construction of coherent states for hamiltonians whose spectrum consists of an equidistant part and one separately disposed level with an energy gap equal to the k skipped levels

variations on a theme of heisenberg pauli and weyl the parentage between weyl pairs the generalized pauli group and the unitary group is investigated in detail we start from an abstract definition of the heisenberg weyl group on the field and then switch to the discrete heisenberg weyl group or generalized pauli group on a finite ring the main characteristics of the latter group an abstract group of order d3 noted pd are given conjugacy classes and irreducible representation classes or equivalently lie algebra of dimension d3 associated with pd leaving the abstract sector a set of weyl pairs in dimension d is derived from a polar decomposition of su 2 closely connected to angular momentum theory then a realization of the generalized pauli group pd and the construction of generalized pauli matrices in dimension d are revisited in terms of weyl pairs finally the lie algebra of the unitary group u d is obtained as a subalgebra of the lie algebra associated with pd this leads to a development of the lie algebra of u d in a basis consisting of d2 generalized pauli matrices in the case where d is a power of a prime integer the lie algebra of su d can be decomposed into d 1 cartan subalgebras

distorted heisenberg algebra and coherent states for isospectral oscillator hamiltonians the dynamical algebra associated with a family of isospectral oscillator hamiltonians is studied through the analysis of its representation in the basis of energy eigenstates it is shown that this representation becomes similar to that of the standard heisenberg algebra and it is dependent on a parameter omega or 0 we call it the distorted heisenberg algebra where omega is the distortion parameter the corresponding coherent states for an arbitrary omega are derived and some particular examples are discussed in detail a prescription to produce the squeezing by adequately selecting the initial state of the system is given

temporally stable coherent states for infinite well and poschl teller potentials this article is a direct illustration of a construction of coherent states which has been recently proposed by two of us jpg and jk we have chosen the example of a particle trapped in an infinite square well and also in poschl teller potentials of the trigonometric type in the construction of the corresponding coherent states we take advantage of the simplicity of the solutions which ultimately stems from the fact they share a common su 1 1 symmetry a la barut girardello many properties of these states are then studied both from mathematical and from physical points of view

phase operators phase states and vector phase states for su3 and su2 1 this paper focuses on phase operators phase states and vector phase states for the sl3 lie algebra we introduce a one parameter generalized oscillator algebra a 2 which provides a unified scheme for dealing with su3 for 0 and h4 h4 for 0 symmetries finite and infinite dimensional representations of a 2 are constructed for 0 and 0 respectively phase operators associated with a 2 are defined and temporally stable phase states as well as vector phase states are constructed as eigenstates of these operators finally we discuss a relation between quantized phase states and a quadratic discrete fourier transform and show how to use these states for constructing mutually unbiased bases

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

higher order susy linearized nonlinear heisenberg algebras and coherent states using an iterative construction of the first order intertwining technique we find k parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below e0 it is seen that the natural ladder operators for these systems give place to polynomial nonlinear algebras and it is shown that these algebras can be linearized the coherent states construction is performed in the nonlinear and linearized cases

generalized coherent and intelligent states for exact solvable quantum systems the so called gazeau klauder and perelomov coherent states are introduced for an arbitrary quantum system we give also the general framework to construct the generalized intelligent states which minimize the robertson schrodinger uncertainty relation as illustration the poschl teller potentials of trigonometric type will be chosen we show the advantage of the analytical representations of gazeau klauder and perelomov coherent states in obtaining the generalized intelligent states in analytical way

topology optimization using a mixed formulation an alternative way to solve pressure load problems abstract a shortcoming of the traditional density based approach to topology optimization is the handling of design dependent loads that relate to boundary data such as for example pressure loads previous works have introduced spline and iso density curves or alternative parametrization schemes to determine the load surfaces in this work we suggest a new way to solve pressure load problems in topology optimization using a mixed displacement pressure formulation for the underlying finite element problem we define the void phase to be an incompressible hydrostatic fluid in this way we can transfer pressure loads through the fluid without any needs for special load surface parametrizations the method is easily implemented in the standard density approach and is demonstrated to work efficiently for both 2d and 3d problems by extending the method to a three phase solid fluid void design method we also demonstrate design of water containing dams

a mimetic discretization of the stokes problem with selected edge bubbles a new mimetic finite difference method for the stokes problem is proposed and analyzed the mimetic discretization methodology can be understood as a generalization of the finite element method to meshes with general polygons polyhedrons in this paper the mimetic generalization of the unstable p 1 p 0 and the conditionally stable q1 p0 finite element is shown to be fully stable when applied to a large range of polygonal meshes moreover we show how to stabilize the remaining cases by adding a small number of bubble functions to selected mesh edges a simple strategy for selecting such edges is proposed and verified with numerical experiments

a c 1 virtual element method for the cahn hilliard equation with polygonal meshes in this paper we develop an evolution of the c 1 virtual elements of minimal degree for the approximation of the cahn hilliard equation the proposed method has the advantage of being conforming in h 2 and making use of a very simple set of degrees of freedom namely 3 degrees of freedom per vertex of the mesh moreover although the present method is new also on triangles it can make use of general polygonal meshes as a theoretical and practical support we prove the convergence of the semidiscrete scheme and investigate the performance of the fully discrete scheme through a set of numerical tests

polygonal finite elements for topology optimization a unifying paradigm in topology optimization literature the parameterization of design is commonly carried out on uniform grids consisting of lagrangian type finite elements e g linear quads these formulations however suffer from numerical anomalies such as checkerboard patterns and one node connections which has prompted extensive research on these topics a problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members leading to mesh dependent sub optimal designs thus to address the geometric features of the spatial discretization we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions more specifically we consider polygonal meshes constructed from voronoi tessellations which in addition to possessing higher degree of geometric isotropy allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities copyright 2009 john wiley sons ltd

topology optimization for stationary fluid structure interaction problems using a new monolithic formulation this paper outlines a new procedure for topology optimization in the steady state fluid structure interaction fsi problem a review of current topology optimization methods highlights the difficulties in alternating between the two distinct sets of governing equations for fluid and structure dynamics hereafter the fluid and structural equations respectively and in imposing coupling boundary conditions between the separated fluid and solid domains to overcome these difficulties we propose an alternative monolithic procedure employing a unified domain rather than separated domains which is not computationally efficient in the proposed analysis procedure the spatial differential operator of the fluid and structural equations for a deformed configuration is transformed into that for an undeformed configuration with the help of the deformation gradient tensor for the coupling boundary conditions the divergence of the pressure and the darcy damping force are inserted into the solid and fluid equations respectively the proposed method is validated in several benchmark analysis problems topology optimization in the fsi problem is then made possible by interpolating young s modulus the fluid pressure of the modified solid equation and the inverse permeability from the damping force with respect to the design variables

toward a stress based topology optimization procedure with indirect calculation of internal finite element information abstract in this work a novel computational approach is developed for the gradient based stress based topology optimization method where the volume is minimized according to locally defined stress constraints of static failure criteria in the framework of tailored finite element fe software without direct access to an internal finite element information database tailored finite element codes that require substantial understanding and modification reprogramming or do not directly provide internal finite element information have rarely been used for stress based topology optimization with solid isotropic material with penalization simp methods therefore much research has been confined to two dimensional problems with rectilinear basic finite elements as complex three dimensional geometries with advanced finite element formulations are not supported to overcome this problem we developed a new computational procedure for sensitivity analysis without direct access to an internal information database a task that has previously been regarded as almost impossible since the calculation of linear strain displacement matrices is required in the sensitivity analysis the present procedure includes a node selection algorithm to efficiently calculate the matrices of irregular shaped finite elements by displacement perturbations the benefits of the present approach are that well established powerful finite element codes i e commercial or sophisticated public fe codes can be easily incorporated for linear stress based topology optimization and any type of finite element formulation can be readily employed in contrast to classical sensitivity analysis small computations are used for sensitivity analysis in this work to demonstrate the validity and efficiency of the present procedure and approach several topology optimization problems with 3d and shell elements are solved

coherent states for transparent potentials the darboux transformation operator method is applied to the investigation of coherent states of transparent multisoliton potentials an isometric correspondence between hilbert spaces of the states of a free particle and a particle moving in the soliton potential is established it is shown that the darboux transformation operator being unbounded and closed cannot realize an isometric mapping between hilbert spaces a quasispectral representation of transformation operators in terms of continuous basis sets is obtained different families of coherent states of the multisoliton potential are introduced measures that realize the resolution of the unity in terms of the projectors on the coherent states vectors are calculated these measures are defined by ordinary smooth functions for the states obtained with the help of bounded transformation operators and by generalized ones otherwise

factorization of analytic representations in the unit disc and number phase statistics of a quantum harmonic oscillator the inner outer part factorization of analytic representations in the unit disc is used for an effective characterization of the number phase statistical properties of a quantum harmonic oscillator it is shown that the factorization is intimately connected with the number phase weyl semigroup and its properties in the barut girardello analytic representation the factorization is implemented as a convolution several examples are given which demonstrate the physical significance of the factorization and its role for quantum statistics in particular we study the effect of phase space interference on the factorization properties of a superposition state

coherent states for systems with discrete and continuous spectrum for a system with one degree of freedom coherent states that are parametrized by classical canonical action angle variables are introduced these states also possess continuity of labelling a resolution of unity and temporal stability the insistence on canonical action angle variables strongly restricts any remaining arbitrariness in the coherent state definition such states are introduced for semibounded hamiltonian operators having either a discrete or a continuous spectrum hamiltonians that have both discrete and continuous parts in their spectrum are also discussed

topology optimization with mixed finite elements on regular grids abstract recently new families of mixed finite elements have been proposed to address the analysis of linear elastic bodies on regular grids adopting a limited number of degrees of freedom per element a two dimensional mixed discretization is implemented to formulate an alternative topology optimization problem where stresses play the role of main variables and both compressible and incompressible materials can be dealt with the structural compliance is computed through the evaluation of the complementary energy whereas the enforcement of stress constraints is straightforward numerical simulations investigate the features of the proposed approach comparisons with a conventional displacement based scheme are provided for compressible materials stress constrained solutions for structures made of incompressible media are introduced

bosonic and k fermionic coherent states for a class of polynomial weyl heisenberg algebras the aim of this paper is to construct coherent states la perelomov and la barut girardello for a polynomial weyl heisenberg algebra this generalized weyl heisenberg algebra denoted by depends on r real parameters and is an extension of the one parameter algebra daoud and kibler 2010 j phys a math theor 43 115303 which covers the cases of the su 1 1 algebra for 0 the su 2 algebra for 0 and the h4 ordinary weyl heisenberg algebra for 0 for finite dimensional representations of and where is a truncation of order s of in the sense of pegg barnett a connection is established with k fermionic algebras or quon algebras this connection makes it possible to use generalized grassmann variables for constructing certain coherent states coherent states of the perelomov type are derived for infinite dimensional representations of and for finite dimensional representations of and through a fock bargmann analytical approach based on the use of complex or bosonic variables the same approach is applied for deriving coherent states of the barut girardello type in the case of infinite dimensional representations of in contrast the construction of coherent states la barut girardello for finite dimensional representations of and can be achieved solely at the price of replacing complex variables by generalized grassmann or k fermionic variables some of the results are applied to su 2 su 1 1 and the harmonic oscillator in a truncated or not truncated form this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

equivalent projectors for virtual element methods in the original virtual element space with degree of accuracy k projector operators in the h 1 seminorm onto polynomials of degree k can be easily computed on the other hand projections in the l 2 norm are available only on polynomials of degree k 2 directly from the degrees of freedom here we present a variant of the virtual element method that allows the exact computations of the l 2 projections on all polynomials of degree k the interest of this construction is illustrated with some simple examples including the construction of three dimensional virtual elements the treatment of lower order terms the treatment of the right hand side and the l 2 error estimates

application of nonlinear deformation algebra to a physical system with poschl teller potential nonlinear deformation algebra are realized in a physical system with p schl teller potential the raising and lowering operators satisfying this algebra are constructed from which the eigenproblem of the system can be exactly solved by the operator method the physical meaning of two deforming functions involved in this algebra is also found in addition algebra is obtained naturally and discussions on the coherent state are also made

polynomial heisenberg algebras polynomial deformations of the heisenberg algebra are studied in detail some of their natural realizations are given by the higher order susy partners and not only by those of first order as is already known of the harmonic oscillator for even order polynomials here it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd moreover it will be proved that the general systems ruled by such kinds of algebras in the quadratic and cubic cases involve painleve transcendents of types iv and v respectively

complementarity and phases in su 3 phase operators and phase states are introduced for irreducible representations of the lie algebra using a polar decomposition of ladder operators in contradistinction with it is found that the polar decomposition does not uniquely determine a hermitian phase operator we describe two possible ways of proceeding one based on imposing su 2 invariance and the other based on the idea of complementarity the generalization of these results to su n is sketched this article is part of a special issue of journal of physics a mathematical and theoretical devoted to coherent states mathematical and physical aspects

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

phase operators temporally stable phase states mutually unbiased bases and exactly solvable quantum systems we introduce a one parameter generalized oscillator algebra that covers the case of the harmonic oscillator algebra and discuss its finite and infinite dimensional representations according to the sign of the parameter we define an hamiltonian operator associated with and examine the degeneracies of its spectrum for the finite when 0 and the infinite when 0 representations of we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators to overcome the difficulties related to the phase operator in the infinite dimensional case and to avoid the degeneracy problem for the finite dimensional case we introduce a truncation procedure which generalizes the one used by pegg and barnett for the harmonic oscillator this yields a truncated generalized oscillator algebra where s denotes the truncation order we construct two types of temporally stable states for as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of two applications are considered in this paper the first concerns physical realizations of and in the context of one dimensional quantum systems with finite morse system or infinite poschl teller system discrete spectra the second deals with mutually unbiased bases used in quantum information

virtual elements for linear elasticity problems we discuss the application of virtual elements to linear elasticity problems for both the compressible and the nearly incompressible case virtual elements are very close to mimetic finite differences see for linear elasticity l beirao da veiga m2an math model numer anal 44 2010 pp 231 250 and in particular to higher order mimetic finite differences as such they share the good features of being able to represent in an exact way certain physical properties conservation incompressibility etc and of being applicable in very general geometries the advantage of virtual elements is the ductility that easily allows high order accuracy and high order continuity

generalized grassmann variables for quantum kit k level systems and barut girardello coherent states for su r 1 algebras this paper concerns the construction of su r 1 barut girardello coherent states in terms of generalized grassmann variables we first introduce a generalized weyl heisenberg algebra a r r 1 generated by r pairs of creation and annihilation operators this algebra provides a useful framework to describe qubit and qukit k level systems it includes the usual weyl heisenberg and su 2 algebras we investigate the corresponding fock representation space the generalized grassmann variables are introduced as variables spanning the fock bargmann space associated with the algebra a r the barut girardello coherent states for su r 1 algebras are explicitly derived and their over completion properties are discussed