on computational shells with scale effects abstract the paper deals with the computations of so called geometrically exact shells with scale effects these type of computations are relevant to thin walled structures at small scales e g thin films nanotubes etc the shell formulation exhibits higher order gradients and is developed following some author s recent work on generalised continua the framework has been modified as to account for two dimensional surfaces as well the shell formulation reduces to a standard one upon disregarding the higher gradient terms the classical part of the formulation is a well known 7 parameter model previously developed by one of the authors which takes thickness change into account the numerical treatment is based on a meshfree formulation which provides the necessary c 1 continuity as possible applications dynamic buckling of cylindrical and spherical shells is investigated and some results for various states of loading and scale parameters are presented the examples include also nanotubes of different dimensions where scale effects matter the influence of the extra gradients is well demonstrated resulting in different buckling behaviours

nonlinear manifold learning for meshfree finite deformation thin shell analysis calculations on general point set surfaces are attractive because of their flexibility and simplicity in the preprocessing but present important challenges the absence of a mesh makes it nontrivial to decide if two neighboring points in the three dimensional embedding are nearby or rather far apart on the manifold furthermore the topology of surfaces is generally not that of an open two dimensional set ruling out global parametrizations we propose a general and simple numerical method analogous to the mathematical theory of manifolds in which the point set surface is described by a set of overlapping charts forming a complete atlas we proceed in four steps 1 partitioning of the node set into subregions of trivial topology 2 automatic detection of the geometric structure of the surface patches by nonlinear dimensionality reduction methods 3 parametrization of the surface using smooth meshfree here maximum entropy approximants and 4 gluing together the patch representations by means of a partition of unity each patch may be viewed as a meshfree macro element we exemplify the generality flexibility and accuracy of the proposed approach by numerically approximating the geometrically nonlinear kirchhoff love theory of thin shells we analyze standard benchmark tests as well as point set surfaces of complex geometry and topology

character generators for elements of finite order in simple lie groups a1 a2 a3 b2 and g2 elements of finite order in the compact simple lie groups su 2 su 3 su 4 sp 4 or o 5 and g 2 are considered we provide the characters of the elements on irreducible representations of lie groups by assigning appro priate numerical values to the variables on which the characters of representations of the lie group depend in this way we specialize generating functions for the characters of the representations of the lie groups to the generating functions for characters of the elements of finite order particular attention is paid to rational elements all of whose characters are integers they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read gaussian elements are also studied in detail their characters are complex valued with integer real and imaginary parts

exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

element formulation and numerical techniques for stability problems in shells in the context of instability problems in shells or shell like structures the objective of the present paper is twofold primarily the paper describes how quasi static conservative instability problems can be considered in a multi parametric context where generalized path following procedures for augmented equilibrium problems are used as computational tools these allow systematic treatment of the higher dimensional solution sets generated under the variations of certain parameters deemed relevant for the given problem the efficient implementation of the above mentioned procedures requires however as an essential ingredient a non linear finite element which is not only accurate but also inexpensive to this end a systematic view on a corotational total lagrangian formulation is described the tric element of argyris and coworkers is slightly modified and introduced as core element formulation special emphasis is given to the alternative methods for treatment of finite three dimensional rotations with reference to both the element definition and solution algorithms numerical examples verify the element capabilities and the possibility to completely describe instability phenomena of large discretized models

the finite deformation theory for beam plate and shell part v the shell element with drilling degree of freedom based on biot strain due to the facts that the spatial moment force is non conservative and the three components of the finite rotation pseudo vector are not independent the ordinary variational principle based on a functional achieving its stationary value is not suitable for the problem subject to non conservative loading the finite element method proposed in this paper is based on the virtual work principle rigid normal assumption polar decomposition and total lagrangian formulation

grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

fast recursion formula for weight multiplicities the purpose of this note is to describe and prove a fast recursion formula for computing multiplicities of weights of finite dimensional representations of simple lie algebras over c until now information about weight multiplicities for all but some special cases 1 2 has had to be found from the recursion formulas of freudenthal 3 or racah 4 typically these formulas become too laborious to use for hand computations for ranks 5 and dimensions 1 0 0 and for ranks 10 and dimensions 10 on a large computer 5 6 with the proposed method the multiplicities can routinely be calculated even by hand for dimensions far exceeding these as an example we present a summary of calculations 7 of all multiplicities in the first sixteen irreducible representations of es let be a semisimple lie algebra over c with root system a and weyl group w relative to a cartan subalgebra let a be the positive roots with respect to some ordering and ii at a the set of simple roots let q and p be the root and weight lattices respectively spanning the real vector space f c if x c p we denote by x the set of dominant elements of x relative to n let m be an irreducible module with highest weight a and weight system 2 an important feature of the approach is the direct determination of 2 without computing outside the dominant chamber since every w orbit is represented by one weight x 2 of the same multiplicity it suffices to compute such x s the recursion formula for computing the multiplicities is a modification proposition 4 of the freudenthal formula in which the weyl group has been exploited to collapse it as much as possible after describing the procedure we present the e example finally the necessary proofs are given

subdivision surfaces a new paradigm for thin shell finite element analysis we develop a new paradigm for thin shell finite element analysis based on the use of subdivision surfaces for i describing the geometry of the shell in its undeformed configuration and ii generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the kirchhoff love theory of thin shells the particular subdivision strategy adopted here is loop s scheme with extensions such as required to account for creases and displacement boundary conditions the displacement fields obtained by subdivision are h2 and consequently have a finite kirchhoff love energy the resulting finite elements contain three nodes and element integrals are computed by a one point quadrature the displacement field of the shell is interpolated from nodal displacements only in particular no nodal rotations are used in the interpolation the interpolation scheme induced by subdivision is non local i e the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighbouring elements however the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface numerical tests including the belytschko et al 10 obstacle course of benchmark problems demonstrate the high accuracy and optimal convergence of the method

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

general charge conjugation operators in simple lie groups a description of particular elements charge conjugation operators found in any compact simple lie group k is presented such elements ri transform a physical state weight vector of a basis of a representation space into others with opposite charge ith component of the weight sometime changing also the sign of the state it is demonstrated that exploitation of these elements and the finite subgroup n of k generated by them offer new powerful methods for computing with representations of the lie group their application to construction of bases in representation spaces is considered in detail it represents a completely new direction to the problem

an algorithm for computing weight multiplicities in irreducible modules for complex semisimple lie algebras abstract let g be a finite dimensional semisimple lie algebra over c having rank l and let v be an irreducible finite dimensional g module having highest weight computations of weight multiplicities in v usually based on freudenthal s formula are in general difficult to carry out in large ranks or for with large coefficients in terms of the fundamental weights in this paper we first show that in some situations these coefficients can be lowered in order to simplify the calculations we then investigate how this can be used to improve the aforementioned formula of freudenthal leading to a more efficient version of the latter in terms of complexity as well as to a way of dealing with certain computations in unbounded ranks we conclude by illustrating the last assertion with a concrete example

green currents for holomorphic automorphisms of compact kahler manifolds and some additional regularity properties very likely the currents t will describe the distribution of invariant manifolds of codimension s corresponding to the small est lyapounov exponents let dp denote the dynamical degree of order p of it describes the growth under iteration of the volume of p dimensional manifolds when d 1 it is natural to introduce first a positive closed 1 l current as ifnyoj 7 lim

on the parametrization of finite rotations in computational mechanics a classification of concepts with application to smooth shells this paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure category ii is based on total rotational degrees of freedom leading to an additive update structure whereas category i relies on linearized rotational degrees of freedom leading to a multiplicative update structure based on this classification new formulations are developed among them are the rodrigues formula applied within category ii with additive update structure for the two components of the rotation vector and two successive elementary rotations in the sense of euler angles applied within category i yielding a singularity free formulation the numerical examples confirm that every rotational parametrization when applied within category ii leads to singularities whereas the application within category i enables the calculation of overall rotations unrestricted in size without any singularity

reduction of orbits of finite coxeter groups of non crystallographic type a reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices which transform points of the orbit of any group into points of the orbits of its subgroup projection matrices and branching rules for orbits of finite coxeter groups of non crystallographic type are presented the novelty in this paper is producing the branching rules that involve non crystallographic coxeter groups moreover these branching rules are relevant to any application of non crystallographic coxeter groups including molecular crystallography and encryption

the signatures of finite dimensional representations of the de sitter groups so and so the signature of a finite dimensional orthogonal representation of a simple lie group is the difference between the number of positive and negative signs in the diagonal form of its symmetric bilinear invariant we derive the expressions for the signatures of all finite dimensional representations of the de sitter groups so 4 1 and so 3 2 in two ways one by means of character values of appropriate elements of adjoint order two on the representation and the other through generating functions

clebsch gordan coefficients for su 5 su 3 su 2 u 1 theories the clebsch gordan coefficients are calculated for the following tensor products of su 5 representations 5 5 5 10 5 10 15 5 15 5 10 10 5 5 10 10 5 24 and 10 24 each case is calculated twice once in a weight vector basis independent of any semisimple subgroup the second time in a basis which refers to su 3 su 2 u 1 su 5

stress resultant geometrically exact form of classical shell model and vector like parameterization of constrained finite rotations summary in this work we consider the geometrically exact shell model subjected to 7nite rotations making use of rotation vector parameters for handling the corresponding constrained rotation for smooth shells a modi7cation of such a parameterization which is based on the incremental rotation vector and thus capable of avoiding the singularity problem is also discussed copyright 2001 john wiley sons ltd

physical stabilization of the 4 node shell element with one point quadrature abstract a new four node shell element for nonlinear analysis which is useful for explicit time integration with single point quadrature is presented an assumed strain method is used to stabilize the zero energy modes of the element the stabilization procedure is based on the hu washizu variational principle and does not require the input of any stabilization parameters example problems show good accuracy and rate of convergence compared with fully integrated 4 node elements however the physically stabilized element is substantially faster than the fully integrated element

a discontinuous galerkin formulation of non linear kirchhoff love shells discontinuous galerkin dg methods provide a means of weakly enforcing the continuity of the unknown field derivatives and have particular appeal in problems involving high order derivatives this feature has previously been successfully exploited comput methods appl mech eng 2008 197 2901 2929 to develop a formulation of linear kirchhoff love shells considering only the membrane and bending responses in this proposed one field method the displacements are the only unknowns while the displacement field is continuous the continuity in the displacement derivative between two elements is weakly enforced by recourse to a dg formulation it is the purpose of the present paper to extend this formulation to finite deformations and non linear elastic behaviors while the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple stress from the displacement field at the mid surface of the shell the non linear formulation considered implies the evaluation of the general stress tensor across the shell thickness leading to a reformulation of the internal forces of the shell nevertheless since the interface terms resulting from the discontinuous galerkin method involve only the resultant couple stress at the edges of the shells the extension to non linear deformations is straightforward copyright 2008 john wiley sons ltd

a finite strain quadrilateral shell element based on discrete kirchhoff love constraints this paper improves the 16 degrees of freedom quadrilateral shell element based on pointwise kirchhoff love constraints and introduces a consistent large strain formulation for this element the model is based on classical shell kinematics combined with continuum constitutive laws the resulting element is valid for large rotations and displacements the degrees of freedom are the displacements at the corner nodes and one rotation at each mid side node the formulation is free of enhancements it is almost fully integrated and is found to be immune to locking or unstable modes the patch test is satisfied in addition the formulation is simple and amenable to efficient incorporation in large scale codes as no internal degrees of freedom are employed and the overall calculations are very efficient results are presented for linear and non linear problems copyright 2005 john wiley sons ltd

a one field full discontinuous galerkin method for kirchhoff love shells applied to fracture mechanics abstract in order to model fracture the cohesive zone method can be coupled in a very efficient way with the finite element method nevertheless there are some drawbacks with the classical insertion of cohesive elements it is well known that on one the hand if these elements are present before fracture there is a modification of the structure stiffness and that on the other hand their insertion during the simulation requires very complex implementation especially with parallel codes these drawbacks can be avoided by combining the cohesive method with the use of a discontinuous galerkin formulation in such a formulation all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements the recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture the purpose of this paper is to develop this formulation for kirchhoff love plates and shells it is achieved by the establishment of a full dg formulation of shell combined with a cohesive model which is adapted to the special thickness discretization of the shell formulation in fact this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations finally numerical examples demonstrate the efficiency of the method

a new discontinuous galerkin method for kirchhoff love shells abstract discontinuous galerkin methods dg have particular appeal in problems involving high order derivatives since they provide a means of weakly enforcing the continuity of the unknown field derivatives this paper proposes a new discontinuous galerkin method for kirchhoff love shells considering only the membrane and bending response the proposed one field method utilizes the weak enforcement in such a way that the displacements are the only unknowns while the rotation continuity is weakly enforced this work presents the formulation of the new discontinuous galerkin method for linear elastic shells demonstrates the consistency and stability of the proposed framework and establishes the method s convergence rate after a description of the formulation implementation into a finite element code these properties are demonstrated on numerical applications

orbit orbit branching rules between simple low rank algebras and equal rank subalgebras complete orbit orbit branching rules or equivalently reduction of weyl group orbits between each simple algebra of rank 3 or less and its equal rank subalgebras and between f4 and each of its equal rank subalgebras are given the generic case an an 1 u 1 the subjoining f4 b3 u 1 and e8 e6 a2 first and seventh highest weight labels nonzero are also treated branching rules between rank 4 and 5 simple algebras and e6 and their equal rank subalgebras are available in a depository

equidistribution speed for endomorphisms of projective spaces let f be a non invertible holomorphic endomorphism of mathbb p k f n its iterate of order n and the equilibrium measure of f we estimate the speed of convergence in the following known result if a is a zariski generic point in mathbb p k the probability measures equidistributed on the preimages of a under f n converge to as n goes to infinity

an explicit discontinuous galerkin method for non linear solid dynamics formulation parallel implementation and scalability properties an explicit dynamics spatially discontinuous galerkin dg formulation for non linear solid dynamics is proposed and implemented for parallel computation dg methods have particular appeal in problems involving complex material response e g non local behavior and failure as even in the presence of discontinuities they provide a rigorous means of ensuring both consistency and stability in the proposed method these are guaranteed the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation the semi discrete system of ordinary differential equations is integrated in time using a conventional second order central difference explicit scheme a stability criterion for the time integration algorithm accounting for the influence of the dg discretization stability is derived for the equivalent linearized system this approach naturally lends itself to efficient parallel implementation the resulting dg computational framework is implemented in three dimensions via specialized interface elements the versatility robustness and scalability of the overall computational approach are all demonstrated in problems involving stress wave propagation and large plastic deformations copyright 2007 john wiley sons ltd

orbit orbit branching rules for families of classical lie algebra subalgebra pairs complete orbit orbit branching rules are derived for each classical algebra maximal subalgebra pair cm ncm cn bm ndm bn dm ndm dn since each pair is equal rank and algebra and subalgebra weyl sectors line up the integrity basis in each case consists of the subalgebra orbits contained in the fundamental orbits of the algebra

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

the demazure tits subgroup of a simple lie group the demazure tits subgroup of a simple lie group g is the group of invariance of clebsch gordan coefficients tables assuming an appropriate choice of basis the structure of the demazure tits subgroups of an bn cn dn and g2 is described orbits of the permutation action of the dt group in any irreducible finite dimensional representation space of a2 c2 and g2 are decomposed into the sum of irreducible representations of the dt group

a discontinuous galerkin method for strain gradient dependent damage study of interpolations and convergence abstract a discontinuous galerkin method has been developed for strain gradient dependent damage the strength of this method lies in the fact that it allows the use of c 0 interpolation functions for continuum theories involving higher order derivatives while in a conventional framework at least c 1 interpolations are required the discontinuous galerkin formulation thereby offers significant potential for engineering computations with strain gradient dependent models when using basis functions with a low degree of continuity jump conditions arise at element edges which are incorporated in the weak form in addition to the formulation itself a detailed study of the convergence properties of the method for various element types is presented and an error analysis is undertaken numerical results of some one and two dimensional problems are presented and discussed

e6 unification model building iii clebsch gordan coefficients in e6 tensor products of the 27 with higher dimensional representations e6 is an attractive group for unification model building however the complexity of a rank 6 group makes it nontrivial to write down the structure of higher dimensional operators in an e6 theory in terms of the states labeled by quantum numbers of the standard model gauge group in this paper we show the results of our computation of the clebsch gordan coefficients for the products of the 27 with irreducible representations of higher dimensionality 78 351 351 351 and 351 application of these results to e6 model building involving higher dimensional operators is straightforward

the state labeling problem a universal solution a new method is proposed for labeling a basis for any irreducible representation with top weight of a simple lie algebra la it has two advantages over gel fand zetlin patterns i being exactly the same for all la s both exceptional and classical and ii providing labels that are much more compact thus it may prove useful in discussing large representations such as those e8 e8 that occur in superstring theory the method makes essential use of the weyl group and is based on a theorem that associates to each weight a symmetric matrix with integer coefficients whose rank equals the multiplicity of the weight

equidistribution speed towards the green current for endomorphisms of pk abstract let f be a non invertible holomorphic endomorphism of p k for a hypersurface h of p k generic in the zariski sense we give an explicit speed of convergence of f n h towards the dynamical green 1 1 current of f

structure of certain weyl modules for the spin groups abstract let k be an algebraically closed field of characteristic p 0 and let w be a finite dimensional k vector space of dimension greater than or equal to 5 in this paper we give the structure of certain weyl modules for g spin w in the case where p 2 as well as the dimension of the corresponding irreducible finite dimensional rational kg modules in addition we determine the composition factors of the restriction of certain irreducible finite dimensional rational k sl w modules to so w

graded contractions and kinematical groups of space time g graded contractions of the complex lie algebra b2 and of its real forms are considered it is found that a particular g z2 z2 is a proper choice of the grading group preserved when graded contractions leading to lie algebras of the kinematical groups of space time are studied

on the reflection type decomposition of the adjoint reduced phase space of a compact semisimple lie group we consider a system with symmetries whose configuration space is a compact lie group acted upon by inner automorphisms the classical reduced phase space of this system decomposes into connected components of orbit type subsets to investigate hypothetical quantum effects of this decomposition one has to construct the associated costratification of the hilbert space of the quantum system in the sense of huebschmann in the present paper instead of the decomposition by orbit types we consider the related decomposition by reflection types conjugacy classes of reflection subgroups these two decompositions turn out to coincide e g for the classical groups su n and sp n we derive defining relations for reflection type subsets in terms of irreducible characters and discuss how to obtain from that the corresponding costratification of the hilbert space of the system to illustrate the method we give explicit results for some low rank classical groups

character calculations for semisimple lie algebras a recursive method for calculation of characters of semisimple lie algebras is outlined by slight modifications algorithms are obtained for the computation of weight multiplicities kronecker products and branching rules as well as symmetrized kronecker powers

local regularity of super potentials and equidistribution of positive closed currents on mathbb p k p k in this paper we introduce and study natural notions of local continuity boundedness of super potentials on mathbb p k next we prove an equidistribution theorem of positive closed p p currents of mathbb p k whose super potentials are continuous bounded near an invariant analytic subset for holomorphic endomorphisms of mathbb p k we also consider the case of regular polynomial automorphisms of mathbb c k

e6 unification model building ii clebsch gordan coefficients of 78 78 we have computed the clebsch gordan coefficients for the product 000 001 000 001 where 000 001 is the adjoint 78 dimensional representation of e6 the results are presented for the dominant weights of the irreducible representations in this product as a simple application we express the singlet operator in 27 78 27 in terms of multiplets of the standard model gauge group

the weyl groups and weight multiplicities of the exceptional lie groups the weyl group wg of each exceptional simple lie group g is described in detail its structure is defined in terms of its coset decomposition with respect to the weyl group wh of a classical semi simple lie group h embedded naturally in g the concepts of g dominance and g equivalence are defined and used to determine from the character formula of weyl the branching rule associated with the restriction of group elements from g to h the weyl group wg is used further to impose constraints on both the branching multiplicities for g to h and the weight multiplicities of g these constraints are used to evaluate the weight multiplicities of f4 e6 e7 and e8 together with the branching multiplicities for e8 to so 16

fine gradings of o 5 c sp 4 c and of their real forms there are three fine gradings of the simple lie algebra of type b2 over the complex number field they provide a basic information about the structure of the algebra in the paper an explicit description of all fine gradings is given in terms of the four dimensional symplectic sp 4 c and five dimensional orthogonal o 5 c representations of the algebra in addition the real forms of b2 are considered it is shown which of the fine gradings survive the restriction to each of the real forms these results should be useful in defining various sets of additive quantum numbers for systems with such symmetries for systematic study of grading preserving contractions of this lie algebra and generally for choosing bases which reflect structural properties of the lie algebra

automorphisms and general charge conjugations it is shown how outer automorphisms of a semisimple lie algebra and automorphisms of the corresponding root system describe conjugations of general charges new results about representations of semidirect products of non abelian lie groups and about projective representations of finite groups are derived

a c0 discontinuous galerkin formulation for kirchhoff plates abstract a particular discontinuous galerkin finite element formulation for the simulation of kirchhoff plates is presented it is rotation free and utilises standard c 0 lagrange finite element basis functions with the required continuity imposed in a weak sense across element boundaries the implications of the scheme in terms of coercivity and convergence of the galerkin problem in various norms are studied with the formulation shown to be stable for any positive value of a penalty parameter a priori error estimates are supported by a range of numerical examples properties of the approach for the important eigenvalue problems of plate buckling and vibration are also examined through numerical examples based on the results of the analysis and numerical examples it is concluded that the formulation is robust accurate and relatively simple

e6 unification model building i clebsch gordan coefficients of 27 27 in an effort to develop tools for grand unified model building for the lie group e6 in this paper we present the computation of the clebsch gordan coefficients for the product 100000 000010 where 100000 is the fundamental 27 dimensional representation of e6 and 000010 is its charged conjugate the results are presented in terms of the dominant weight states of the irreducible representations in this product these results are necessary for the group analysis of e6 operators also involving higher representations which is the next step in this project in this paper we apply the results to the construction of the operator 273

clebsch gordan coefficients for su 5 unification models the clebsch gordan coefficients for the product 1001 1001 where 1001 is the adjoint representation of su 5 with respect to the group basis and the subgroup basis in the reduction su 5 su 3 su 2 u 1 are computed one of the basic tools in this computation is the exhaustive use of the verma algorithm to find bases for the weight subspaces of dimension higher than 1 it allows for the construction of bases in a systematic way by using the so called verma inequalities only the coefficients for the dominant weights are calculated the other ones can be obtained by using the elements of finite order charge conjugation operators of su 5

physical relations and the weyl group let a given particle symmetry be described by a reductive lie group g it is proved that the corresponding weyl group w r acts canonically in all zero weight spaces of g and hence in particular on observables moreover it is shown how this w r action provides many physical relations including those believed to be implied by g transformation properties of observables the results simplify a testing of symmetries based on various lie groups algebras their use extends beyond particle physics e g to nuclear physics

geometrically nonlinear formulation for thin shells without rotation degrees of freedom abstract a finite element formulation is presented for modelling geometrically nonlinear thin shells which exploits standard lagrange finite element basis functions without introducing rotation degrees of freedom the classical regularity requirements associated with thin bending problems are circumvented by introducing special integrals over inter element edges the use of lagrange finite element basis functions and the absence of rotation degrees of freedom make the formulation relatively simple and discontinuities in material properties and non smooth shell geometry can be incorporated trivially the variational problem can be exactly linearised leading to an efficient newton raphson solution process the performance of the approach is demonstrated via a range of numerical benchmarks for both geometrically linear and nonlinear problems it is shown that cubic elements perform particularly well

on orbit sum values of elements of finite order the irreducible characters form a basis of the ring of class functions on a compact simply connected simple lie group another basis are the orbit sums f l s m w le m which are related to the irreducible characters by the inner multiplicities there is a w invariant sesquilinear form on the class functions defined over a finite set of elements of finite order under which the orbit sums are almost orthogonal we give an exact criterion for the orthogonality given the set of elements of order dividing a fixed n and follow some remarkable properties of the values of orbit sums on elements of finite order efo

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

multiplicity of holomorphic functions the main result of this paper concerns an analytic version of birkhoff s ergodic theorem it allows us to define asymptotic multiplicities given any algebraically stable rational map f mathbb p 2 to mathbb p 2 this is the key for our study of the singularities of the green current t associated to f we characterize the points where the lelong number of t is strictly positive

quantum r matrix for e7 and f4 groups the quantum r matrix for e7 and f4 are explicitly constructed for the defining representations 56 and 26 of e7 and f4 respectively for these skein relations link polynomials and spectral parameter dependent solutions of yang baxter equation are obtained the r matrix for the matrix elements related by the weyl reflection has different values surprisingly

super potentials of positive closed currents intersection theory and dynamics we introduce a notion of super potential for positive closed currents of bidegree p p on projective spaces this gives a calculus on positive closed currents of arbitrary bidegree we define in particular the intersection of such currents and the pull back operator by meromorphic maps one of the main tools is the introduction of structural discs in the space of positive closed currents which gives a geometry on that space we apply the theory of super potentials to construct green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms

a four node plate bending element based on mindlin reissner plate theory and a mixed interpolation this communication discusses a 4 node plate bending element for linear elastic analysis which is obtained as a special case from a general nonlinear continuum mechanics based 4 node shell element formulation the formulation of the plate element is presented and the results of various example solutions are given that yield insight into the predictive capability of the plate and shell element

geometrical classification scheme for weights of lie algebras every weight of a lie algebra belongs to one particular weyl orbit the depth within an orbit has been defined as the minimum number of weyl reflections associated with simple roots that are necessary to transform the weight into the dominant weight of the orbit a simple set of classification parameters that measures the depth and distinguishes between different weights of the same depth is introduced the parameters are geometrical in that they are related directly to the orientation of the weight vector they correspond to a particular member of the weyl group that transforms the weight into the dominant weight a simple rule is given for listing all the weights of an orbit in terms of the parameters

discontinuous galerkin methods for non linear elasticity this paper presents the formulation and a partial analysis of a class of discontinuous galerkin methods for quasistatic non linear elasticity problems these methods are endowed with several salient features the equations that define the numerical scheme are the euler lagrange equations of a one field variational principle a trait that provides an elegant and simple derivation of the method in consonance with general discontinuous galerkin formulations it is possible within this framework to choose different numerical fluxes numerical evidence suggests the absence of locking at near incompressible conditions in the finite deformations regime when piecewise linear elements are adopted finally a conceivable surprising characteristic is that as demonstrated with numerical examples these methods provide a given accuracy level for a comparable and often lower computational cost than conforming formulations stabilization is occasionally needed for discontinuous galerkin methods in linear elliptic problems in this paper we propose a sufficient condition for the stability of each linearized non linear elastic problem that naturally includes material and geometric parameters the latter needed to account for buckling we then prove that when a similar condition is satisfied by the discrete problem the method provides stable linearized deformed configurations upon the addition of a standard stabilization term we conclude by discussing the complexity of the implementation and propose a computationally efficient approach that avoids looping over both elements and element faces several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous galerkin method within the class copyright 2006 john wiley sons ltd

general indices of representations and casimir invariants a modified definition of the index of degree p of a finite dimensional representation of a simple lie algebra is given the definition applies equally to even and odd p the correspondence between the earlier definitions of indices even p and anomaly numbers odd p is pointed out as well as the relation to casimir invariants of the algebra a closed formula for the fifth order index is derived

clebsch gordan coefficients for e6 and so 10 unification models we illustrate here a new method for computing clebsch gordan coefficients cgc for e6 by computing cgc for the product 27 27 of the irreducible representation 100000 of e6 with itself these cgc are calculated thrice once in a weight vector basis independent of any semisimple subgroup then in a basis which refers to so 10 e6 and finally in a basis referring to su 5 so 10 e6

verma bases for representations of classical simple lie algebras complete bases are constructed for all finite dimensional irreducible representations of the simple lie algebras over c of the types an n 1 bn and cn 2 n 6 dn 4 n 6 and g2 each basis vector is given as an explicit sequence of weight lowering generators of the algebra acting on the highest weight vector of the representation space a similar construction due to d n verma for the highest weight representations of all kac moody algebras of rank 2 is presented as well

generating functions and multiplicity formulas the case of rank two simple lie algebras a procedure is described that makes use of the generating function of characters to obtain a new generating function h giving the multiplicities of each weight in all the representations of a simple lie algebra the way to extract from h explicit multiplicity formulas for particular weights is explained and the results corresponding to rank two simple lie algebras are shown

the weyl orbits of g2 f4 e6 and e7 simple methods are described for constructing the weyl orbits of exceptional lie algebras and for identifying the orbit of an arbitrary weight the methods make use of convenient bases in weight space they are applied to g2 f4 e6 and e7 a complete table of all weights of all orbits of f4 is given the weights of the 37 orbits of smallest weight length of e6 and e7 are given in tabular form the depth structures of the orbits are discussed

a discontinuous galerkin formulation for a strain gradient dependent damage model abstract the numerical solution of strain gradient dependent continuum problems has been dogged by continuity demands on the basis functions for most commonly accepted models solutions using the finite element method demand c 1 continuity of the shape functions here recent developments in discontinuous galerkin methods are explored and exploited for the solution of a prototype non linear strain gradient dependent continuum model a formulation is developed that allows the rigorous solution of a strain gradient damage model using standard c 0 shape functions the formulation is tested in one dimension for the simplest possible finite element formulation continuous piecewise linear displacement and constant on elements internal variable numerical results are shown to compare excellently with a benchmark solution the results are remarkable given the simplicity of the proposed formulation

a general discontinuous galerkin method for finite hyperelasticity formulation and numerical applications a discontinuous galerkin formulation of the boundary value problem of finite deformation elasticity is presented the primary purpose is to establish a discontinuous galerkin framework for large deformations of solids in the context of statics and simple material behaviour with a view toward further developments involving behaviour or models where the dg concept can show its superiority compared to the continuous formulation the method is based on a general hu washizu de veubeke functional allowing for displacement and stress discontinuities in the domain interior it is shown that this approach naturally leads to the formulation of average stress fluxes at interelement boundaries in a finite element implementation the consistency and linearized stability of the method in the non linear range as well as its convergence rate are proven an implementation in three dimensions is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward manner in order to demonstrate the versatility accuracy and robustness of the method examples of application and convergence studies in three dimensions are provided copyright 2006 john wiley sons ltd

graded contractions of jordan algebras and of their representations contractions of jordan algebras and jordan superalgebras which preserve a chosen grading are defined and studied simultaneous grading of jordan algebras and their representation spaces is used to develop a theory of grading preserving contractions of representations of jordan algebras

the rings of n dimensional polytopes points of an orbit of a finite coxeter group g generated by n reflections starting from a single seed point are considered as vertices of a polytope g polytope centered at the origin of a real n dimensional euclidean space a general efficient method is recalled for the geometric description of g polytopes their faces of all dimensions and their adjacencies products and symmetrized powers of g polytopes are introduced and their decomposition into the sums of g polytopes is described several invariants of g polytopes are found namely the analogs of dynkin indices of degrees 2 and 4 anomaly numbers and congruence classes of the polytopes the definitions apply to crystallographic and non crystallographic coxeter groups examples and applications are shown

branching rules for representations of simple lie algebras through weyl group orbit reduction two independent algorithms are presented which together allow the determination of branching rules from an irreducible representation of a compact lie algebra to those of a subalgebra or subjoined algebra the first gives the subalgebra weyl orbits contained in an algebra orbit the second gives the irreducible representations of an algebra contained in an orbit and by inversion of a triangular matrix the orbits contained in an irreducible representation