asymptotic mesh independence of newton s method revisited the paper presents a new affine invariant theory on asymptotic mesh independence of newton s method for discretized nonlinear operator equations compared to earlier attempts the new approach is both much simpler and more intuitive from the algorithmic point of view the theory is exemplified at finite element methods for elliptic pde problems

an objective 3d large deformation finite element formulation for geometrically exact curved kirchhoff rods the objective of this work is the development of a new finite element formulation for beams according to the kirchhoff theory of thin rods which includes the deformation states of axial tension torsion and bending the proposed formulation accounts for large deformations in three dimensional problem settings consisting of slender prismatic beams with arbitrarily curved initial geometries and arbitrary cross section shapes like in geometrically exact reissner theories the derived deformation measures are geometrically exact in the sense that they are consistent with the strong and weak form of the balance equations and the beam geometry for any configuration and for arbitrarily large translations rotations and strains a novel orthogonal interpolation strategy is applied to the triad field representing the cross section orientation in order to fulfill the kirchhoff constraint of vanishing shear strains in a strong sense and to preserve the objectivity of the spatially discretized problem the continuity requirements resulting from the corresponding weak form are fulfilled by a cubic hermite interpolation thus leading to a c1 continuous representation of the beam centerline fundamental properties such as objectivity path independence consistency and accuracy of the developed beam element are verified by means of suitable numerical examples

on totally real cubic fields the authors have constructed a table of the 26440 nonconjugate totally real cubic number fields of discriminant d 500000 thereby extending the existing table of fields with d 100000 by i 0 angell 1 serious defects in angell s table are pointed out for each field running number discriminant coefficients of a generating polynomial integral basis class number and a fundamental pair of units are listed the article contains statistics about the following subjects distribution of class numbers fields in which every norm positive unit is totally positive nonconjugate fields with the same discriminant fields with noncyclic class group the fields are tabulated by means of a method due to davenport and heilbronn 7 8 which leads to a unique normalized generating polynomial the given units are chosen so that the fundamental parallelogram of the unit lattice determined by the corresponding vectors in the logarithmic space is reduced

geodesic finite elements on simplicial grids summary we introduce geodesic finite elements as a conforming way to discretize partial differential equations for functions v m where is an open subset of rd and m is a riemannian manifold these geodesic finite elements naturally generalize standard first order finite elements for euclidean spaces they also generalize the geodesic finite elements proposed for d 1 in a previous publication of the author our formulation is equivariant under isometries of m and hence preserves objectivity of continuous problem formulations we concentrate on partial differential equations that can be formulated as minimization problems discretization leads to algebraic minimization problems on product manifolds mn these can be solved efficiently using a riemannian trust region method we propose a monotone multigrid method to solve the constrained inner problems with linear multigrid speed as an example we numerically compute harmonic maps from a domain in r3 to s2 copyright 2012 john wiley sons ltd

l 2 discretization error bounds for maps into riemannian manifolds we study the approximation of functions that map a euclidean domain omega subset mathbb r d into an n dimensional riemannian manifold m g minimizing an elliptic semilinear energy in a function set h subset w 1 2 omega m the approximation is given by a restriction of the energy minimization problem to a family of conforming finite dimensional approximations s h subset h we provide a set of conditions on s h such that we can prove a priori w 1 2 and l 2 approximation error estimates comparable to standard euclidean finite elements this is done in an intrinsic framework independently of embeddings of the manifold or the choice of coordinates

a broyden class of quasi newton methods for riemannian optimization this paper develops and analyzes a generalization of the broyden class of quasi newton methods to the problem of minimizing a smooth objective function f on a riemannian manifold a condition on vector transport and retraction that guarantees convergence and facilitates efficient computation is derived experimental evidence is presented demonstrating the value of the extension to the riemannian broyden class through superior performance for some problems compared to existing riemannian bfgs methods in particular those that depend on differentiated retraction

galerkin lie group variational integrators based on unit quaternion interpolation abstract lie group variational integrators of arbitrary order are developed using the galerkin method based on unit quaternion interpolation to our knowledge quaternions have not been used before for this purpose though they allow a very simple and efficient way to perform the interpolation the resulting integrators are symplectic and structure preserving in the sense that certain symmetries in the lagrangian of the mechanical system are carried over to the discrete setting which leads to the preservation of the corresponding momentum maps the integrators furthermore exhibit a very good long time energy behavior i e energy is neither dissipated nor gained artificially at the same time the lie group structure is preserved by carefully defining the variations the interpolation method and by solving the non linear system of equations directly on the manifold rather than constraining it in a surrounding space using lagrange multipliers as a consequence we are able to show that lie group variational integrators based on the special orthogonal group are equivalent to the variational integrators for constrained systems using the discrete null space method employed e g in dmocc discrete mechanics and optimal control of constrained systems we show new numerical results on the convergence rates which are substantially higher than the known theoretical bounds and on the relation between accuracy and computational cost

on the computation of a table of complex cubic fields with discriminant d 10 6 a method for finding all the nonisomorphic complex cubic fields with discriminant d 106 is described three different methods were used to find the class number of each of these fields the speed of these techniques is discussed and several tables illustrating the computational results are presented these include tables of the distribution of the fields and the class numbers and the class group structures of these fields

interpolation in lie groups we consider interpolation in lie groups based on points on the manifold together with tangent vectors at some of these points we construct hermite interpolation polynomials if the points and tangent vectors are produced in the process of integrating an ordinary differential equation in terms of lie algebra actions we use the truncated inverse of the differential of the exponential mapping and the truncated baker campbell hausdorff formula to relatively cheaply construct an interpolation polynomial much effort has lately been put into research on geometric integration i e the process of integrating differential equations in such a way that the configuration space of the true solution is respected by the numerical solution some of these methods may be viewed as generalizations of classical methods and we investigate the construction of intrinsic dense output devices as generalizations of the continuous runge kutta methods

scattered manifold valued data approximation we consider the problem of approximating a function f from an euclidean domain to a manifold m by scattered samples f xi i i in mathcal i f i i i where the data sites xi i i in mathcal i i i i are assumed to be locally close but can otherwise be far apart points scattered throughout the domain we introduce a natural approximant based on combining the moving least square method and the karcher mean we prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart the analysis also tells us that the use of karcher s mean dependent on a riemannian metric and the associated exponential map is inessential and one can replace it by a more general notion of center of mass based on a general retraction on the manifold consequently we can substitute the karcher mean by a more computationally efficient mean we illustrate our work with numerical results which confirm our theoretical findings

a locking free finite element formulation and reduced models for geometrically exact kirchhoff rods in this work we suggest a locking free geometrically exact finite element formulation incorporating the modes of axial tension torsion and bending of thin kirchhoff beams with arbitrary initial curvatures and cross section shapes the proposed formulation has been designed in order to represent general load cases and three dimensional problem settings in the geometrically nonlinear regime of large deformations from this comprehensive theory we not only derive a general beam model but also several reduced formulations which deliver accurate solutions for special problem classes concerning the beam geometry and the external loads the advantages of these reduced models arise for example in terms of simplified finite element formulations less degrees of freedom per element and consequently a higher computational efficiency of the corresponding numerical models a second core topic of this publication is the treatment of membrane locking which is a locking phenomenon predominantly occurring in highly slender curved structures thus exactly in the prime field of application for kirchhoff theories in order to address the membrane locking effect we will propose a new interpolation strategy for the axial strain field and compare this method with common approaches such as assumed natural strains ans or reduced integration the effectiveness of this method as well as the consistency and accuracy of the general finite element formulation and the reduced beam models will be illustrated with selected numerical examples

objectivity of strain measures in the geometrically exact three dimensional beam theory and its finite element implementation the paper discusses the issue of discretization of the strainconfiguration relationships in the geometrically exact theory of threedimensional 3d beams which has been at the heart of most recent

the hasse norm principle for abelian extensions we study the distribution of abelian extensions of bounded discriminant of a number field k which fail the hasse norm principle for example we classify those finite abelian groups g for which a positive proportion of g extensions of k fail the hasse norm principle we obtain a similar classification for the failure of weak approximation for the associated norm one tori these results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed which we achieve using tools from harmonic analysis building on work of wright

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

geodesic finite elements for cosserat rods we introduce geodesic finite elements as a new way to discretize the non linear configuration space of a geometrically exact cosserat rod these geodesic finite elements naturally generalize standard one dimensional finite elements to spaces of functions with values in a riemannian manifold for the special orthogonal group our approach reproduces the interpolation formulas of crisfield and jelenic geodesic finite elements are conforming and lead to objective and path independent problem formulations we introduce geodesic finite elements for general riemannian manifolds discuss the relationship between geodesic finite elements and coefficient vectors and estimate the interpolation error then we use them to find static equilibria of hyperelastic cosserat rods using the riemannian trust region algorithm of absil et al we show numerically that the discretization error depends optimally on the mesh size copyright 2009 john wiley sons ltd

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

on the average number of octahedral newforms of prime level we show that on average the number of octahedral newforms of prime level is bounded by a constant

p ell torsion points in finite abelian groups and combinatorial identities abstract the main aim of this article is to compute all the moments of the number of p l torsion elements in some type of finite abelian groups the averages involved in these moments are those defined for the cohen lenstra heuristics for class groups and their adaptation for tate shafarevich groups in particular we prove that the heuristic model for tate shafarevich groups is compatible with the recent conjecture of poonen and rains about the moments of the orders of p selmer groups of elliptic curves for our purpose we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way moreover from our probabilistic model we derive combinatorial identities some of which appearing to be new the others being related to the theory of symmetric functions in some sense our method therefore gives for these identities a somehow natural algebraic context

theory and numerics of three dimensional beams with elastoplastic material behaviour a theory of space curved beams with arbitrary cross sections and an associated finite element formulation is presented within the present beam theory the reference point the centroid the centre of shear and the loading point are arbitrary points of the cross section the beam strains are based on a kinematic assumption where torsion warping deformation is included each node of the derived finite element possesses seven degrees of freedom the update of the rotational parameters at the finite element nodes is achieved in an additive way applying the isoparametric concept the kinematic quantities are approximated using lagrangian interpolation functions since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load carrying capacity of beam structures copyright 2000 john wiley sons ltd

numerical solution of a bending torsion model for elastic rods aiming at simulating elastic rods we discretize a rod model based on a general theory of hyperelasticity for inextensible and unshearable rods after reviewing this model and discussing topological effects of periodic rods we prove convergence of the discretized functionals and stability of a corresponding discrete flow our experiments numerically confirm thresholds e g for michell s instability and indicate a complex energy landscape in particular in the presence of impermeability

enumerative galois theory for cubics and quartics abstract we show that there are o e h 1 5 e monic cubic polynomials with integer coefficients bounded by h in absolute value whose galois group is a 3 we also show that the order of magnitude for d 4 quartics is h 2 log h 2 and that the respective counts for a 4 v 4 c 4 are o h 2 91 o h 2 log h o h 2 log h our work establishes that irreducible non s 3 cubic polynomials are less numerous than reducible ones and similarly in the quartic setting these are the first two solved cases of a 1936 conjecture made by van der waerden

geometrically exact thin walled beam including warping formulated on the special euclidean group se 3 abstract based on a formulation on the special euclidean group s e 3 a geometrically exact thin walled beam with an arbitrary open cross section is proposed to deal with the finite deformation and rotation issues the beam strains are based on a kinematic assumption where warping deformation and wagner effects are included such that the nonlinear behavior of a thin walled beam is predicted accurately particular under large torsion to reduce the nonlinearity of rigid motion static and dynamic equations are derived in the s e 3 framework based on the local frame approach as the value of the iteration matrix including the jacobian matrix of inertial and internal forces is invariable under arbitrary rigid motion the number of updates required during the computation process decreases sharply which drastically improves the computational efficiency furthermore the isogeometric analysis iga based on the non uniform rational b splines nurbs basis functions which promotes the integration of computer aided design cad and computer aided engineering cae is adopted to interpolate the displacement rotation and warping fields separately the interpolated strain measures satisfy the objectivity by removing the rigid motion of the reference point to obtain the symmetric jacobian matrix of internal forces the linearization operation is conducted based on the previously converged configuration a lie group s e 3 extension of the generalized time integration method is utilized to solve the equations of motion for thin walled beams finally the proposed formulation is successfully tested and validated in several static and dynamic numerical examples

geometrically exact beam finite element formulated on the special euclidean group se 3 abstract this paper describes a dynamic formulation of a straight beam finite element in the setting of the special euclidean group se 3 first the static and dynamic equilibrium equations are derived in this framework from variational principles then a non linear interpolation formula using the exponential map is introduced it is shown that this framework leads to a natural coupling in the interpolation of the position and rotation variables next the discretized internal and inertia forces are developed the semi discrete equations of motion take the form of a second order ordinary differential equation on a lie group which is solved using a lie group time integration scheme it is remarkable that no parameterization of the nodal variables needs to be introduced and that the proposed lie group framework leads to a compact and easy to implement formulation some important numerical and theoretical aspects leading to a computationally efficient strategy are highlighted and discussed for instance the formulation leads to invariant tangent stiffness and mass matrices under rigid body motions and a locking free element the proposed formulation is successfully tested in several numerical static and dynamic examples

intrinsic representation of tangent vectors and vector transports on matrix manifolds the quasi newton methods on riemannian manifolds proposed thus far do not appear to lend themselves to satisfactory convergence analyses unless they resort to an isometric vector transport this prompts us to propose a computationally tractable isometric vector transport on the stiefel manifold of orthonormal p frames in mathbb r n rn specifically it requires o np 2 o np2 flops which is considerably less expensive than existing alternatives in the frequently encountered case where n gg p n p we then build on this result to also propose computationally tractable isometric vector transports on other manifolds namely the grassmann manifold the fixed rank manifold and the positive semidefinite fixed rank manifold in the process we also propose a convenient way to represent tangent vectors to these manifolds as elements of mathbb r d rd where d is the dimension of the manifold we call this an intrinsic representation as opposed to extrinsic representations as elements of mathbb r w rw where w is the dimension of the embedding space finally we demonstrate the performance of the proposed isometric vector transport in the context of a riemannian quasi newton method applied to minimizing the brockett cost function

mixed models and reduced selective integration displacement models for nonlinear analysis of curved beams simple mixed models are developed for the geometrically nonlinear analysis of deep arches a total lagrangian description of the arch deformation is used and the analytical formulation is based on a form of the nonlinear deep arch theory with the effects of transverse shear deformation included the fundamental unknowns consist of the six internal forces and generalized displacements of the arch and the element characteristic arrays are obtained by using hellinger reissner mixed variational principle the polynomial interpolation functions used in approximating the forces are one degree lower than those used for approximating the displacements and the forces are discontinuous at the interelement boundaries the equivalence between the mixed models developed herein and displacement models based on reduced integration of both the transverse shear and extensional energy terms is discussed the advantages of mixed models over equivalent displacement models are outlined numerical results are presented to demonstrate the high accuracy and effectiveness of the mixed models developed and compare their performance with other mixed models reported in the literature

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

constructing reference metrics on multicube representations of arbitrary manifolds reference metrics are used to define the differential structure on multicube representations of manifolds i e they provide a simple and practical way to define what it means globally for tensor fields and their derivatives to be continuous this paper introduces a general procedure for constructing reference metrics automatically on multicube representations of manifolds with arbitrary topologies the method is tested here by constructing reference metrics for compact orientable two dimensional manifolds with genera between zero and five these metrics are shown to satisfy the gauss bonnet identity numerically to the level of truncation error which converges toward zero as the numerical resolution is increased these reference metrics can be made smoother and more uniform by evolving them with ricci flow this smoothing procedure is tested on the two dimensional reference metrics constructed here these smoothing evolutions using volume normalized ricci flow with deturck gauge fixing are all shown to produce reference metrics with constant scalar curvatures at the level of numerical truncation error

equilibrium states of an elastic conducting rod in a magnetic field our principal concern is an analysis of the equilibrium states of a nonlinearly elastic conducting rod in a magnetic field we assume hyperelasticity so the equilibria formally appear as critical points of a potential energy functional on the strains fairly standard methods give existence of a minimum not necessarily unique with e g l2 regularity the assumptions imposed on the functional preclude the use of the usual techniques for justification of the formal necessary conditions for optimality a new general technique is developed to justify these conditions it then follows that minimizers satisfy the equilibrium conditions in the classical sense a feature of this technique is that the variations considered are homotopies so one can consider minimization within a homotopy class in the symmetric case which admits trivial straight and untwisted solutions we show that nontrivial solutions also exist if the field is strong enough

on the davenport heilbronn theorems and second order terms we give simple proofs of the davenport heilbronn theorems which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3 torsion elements in the class groups of quadratic fields having bounded discriminant we also establish second main terms for these theorems thus proving a conjecture of roberts our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings