topologies on measured groupoids abstract if an analytic borel group g has a quasiinvariant measure it is known that g is actually a locally compact group with the original borel structure being generated by the topology and the original measure being equivalent to haar measure in this paper a variation is given on the known proof which then extends to show that an analytic measured groupoid has a compact and also a locally compact inessential reduction which is a topological groupoid in the compact case it is proved that every almost homomorphism agrees a e with a strict homomorphism also the topology is used to show that every measured groupoid has a complete countable section 7 and that every locally compact equivalence relation has a complete transversal 3 these are further used to show that some results of feldman et al 7 apply in general and that a locally compact groupoid with continuous haar system has sufficiently many non singular borel g sets provided that the orbit measures are atom free 23

representable e theory for c0 x algebras let x be a locally compact space and let a and b be c0 x algebras we define the notion of an asymptotic c0 x morphism from a to b and construct representable e theory groups re x a b these are the universal groups on the category of separable c0 x algebras that are c0 x stable c0 x homotopy invariant and half exact if a is rkk x nuclear these groups are naturally isomorphic to kasparov s representable kk theory groups rkk x a b applications and examples are also discussed let x be a locally compact space a c algebra a is called a c0 x algebra if it is equipped with a nondegenerate action of c0 x as commuting multipliers of a these algebras have many important applications since they can be represented as the sections of a noncommutative bundle of c algebras over x kw95 ew97 in particular any continuous trace c algebra with spectrum x is a c0 x algebra rw98 there is a substantial amount of literature on c0 x algebras see the general treatments in bla96 nil96 in 1988 kasparov kas88 defined representable kk theory groups rkk x a b which extend his bivariant kk theory groups kk a b where x and the representable topo logical k theory rk 0 x of segal seg70 these groups and their equivariant versions have found many applications in geometry topology and index theory most notably in kasparov s proof of the novikov conjecture for closed discrete subgroups of finite component lie groups kas88 however these groups are difficult to compute sincethey have six term exact sequences only for split exact sequences of c0 x algebras bau98 in 1989 connes and higson ch89 defined e theory groups e a b based on asymptotic morphisms from a to b an asymptotic morphism t a b is a one parameter family of maps which satisfy the properties of a homomorphism in the limit as t there is a natural notion of homotopy for asymptotic morphisms for which a composition product is well defined if a is separable the functor b 7 e a b is the universal functor on the category of separable c algebras b which is stable homotopy invariant and half exact thus there are six term exact sequences in both variables for any exact sequence of separable c algebras there is a natural transformation kk a b e a b which is an isomorphism if a is k nuclear in this paper we extend these constructions to the category of separable c0 x algebras by defining a notion of asymptotic c0 x morphism between two c0 x algebras a and b there is a corresponding notion of c0 x homotopy for an asymptotic c0 x morphism for which composition is well defined this leads us to define representable e theory groups re x a b if x is second countable then these groups have a composition product re x a b re x b c re x a c and we prove that these are the universal such groups on the category of separable c0 x algebras that are c0 x stable c0 x homotopy invariant and half exact we also prove that there is a natural transformation

discontinuity capturing operators for elastodynamics finite element methods are presented which include discontinuity capturing operators to accurately solve elastodynamics problems exhibiting sharp gradients in their solution e g wave propagation the formulation involves finite element discretization of the temporal domain as well as the usual finite element discretization of the spatial domain linear least squares terms are included that enhance the stability of the space time finite element method nonlinear discontinuity capturing operators are presented that result in more accurate capturing of wave fronts in transient solutions while maintaining the high order accuracy of the underlying linear algorithm in smooth regions stability of the discontinuity capturing formulation is proved and a convergence proof is given for a particular choice of the discontinuity capturing operator numerical results are presented that demonstrate the effectiveness of the proposed discontinuity capturing operators for elastodynamics

a new explicit predictor multicorrector high order accurate method for linear elastodynamics abstract a new explicit predictor multicorrector high order accurate method is suggested for linear elastodynamics the method is derived from the implicit high order accurate method based on the time continuous galerkin method proposed earlier in our papers the basic unknowns for the method are displacements and velocities accelerations are not calculated the explicit method uses a predictor multicorrector technique with one or two passes in order to reach the fourth order of accuracy and has controllable numerical dissipation for the suppression of spurious high frequency oscillations in contrast to recently suggested explicit high order accurate methods based on the time discontinuous galerkin method the new method is more accurate has a higher order of accuracy and has better algorithmic properties e g a higher stability limit at the same computational efforts presented numerical examples show the performance of the new method the method appears to be competitive for medium and long term analysis when accuracy of numerical solutions arises an issue due to error accumulation

equivariant e theory for groupoids acting on c algebras we define equivariant asymptotic morphisms between c0 x algebras and construct the category of homotopy classes of such morphisms based on this category we construct the category of homotopy classes of asymptotic morphisms which are equivariant with respect to an action of a groupoid we then define and study the corresponding equivariant e theory

higher derivative explicit one step methods for non linear dynamic problems part i design and theory a new higher order explicit one step scheme is presented for use in the solution of a certain class of non linear problems in structural dynamics the generalized scheme covers some recently developed multistep algorithms as well as some new one step methods with improved numerical properties a weighted residual approach is used which contains new aspects for design of single step algorithms accuracy and stability are investigated for linear and non linear systems algorithms with higher accurary than the central difference method are developed and optimized to minimize the reduction in the stability limit

explicit time integration algorithms for structural dynamics with optimal numerical dissipation abstract a new predictor corrector explicit time integration algorithm is presented for solving structural dynamics problems the basis of the algorithm is the implicit generalized method developed by the authors like its implicit parent the explicit generalized method is a one parameter family of algorithms in which the parameter defines the high frequency numerical dissipation for a given value of high frequency dissipation the explicit generalized method minimizes low frequency dissipation the algorithm can be utilized effectively for structural dynamics calculations in which numerical dissipation is needed to reduce spurious oscillations parameter values are given that enable physical damping to be treated explicitly while maintaining second order accuracy without the need for an expensive second corrector pass

explicit predictor multicorrector time discontinuous galerkin methods for non linear dynamics abstract explicit predictor multicorrector time discontinuous galerkin tdg methods developed for linear structural dynamics are formulated and implemented in a form suitable for arbitrary non linear analysis of structural dynamics problems the formulation is intended to inherit the accuracy properties of the exact parent implicit tdg methods to this end suitable predictors and correctors are designed to achieve third order accuracy large stability limits and controllable numerical dissipation by means of an algorithmic parameter as the study of a general non linear case is rather complex the analysis of the convergence properties of the resulting algorithms are restricted to conservative duffing oscillators for which closed form solutions are available it is shown that the main properties of the underlying parent scheme can be retained finally results of representative numerical simulations relevant to duffing oscillators and to a stiff spring pendulum discretized with finite elements illustrate the performance of the numerical schemes and confirm the analytical estimates

a unified set of single step algorithms part 3 the beta m method a generalization of the newmark scheme introduced herein is a generalization of newmark s time marching integration scheme called the m method like the sspj method introduced in parts 1 and 2 of this series the m method provides a gcneral single step scheme applicable to any set of initial value problems the method is specialized by specifying the method order m along with rn integration parameters 0 1 m 1 for a particular choice of m the integration parameters provide a subfamily of methods which control accuracy and stability as well as offering options for explicit or implicit algorithms for the most part attention is focused on the application to structural dynamic equations most well known methods e g newmark wilson houbolt etc are shown to be special cases within the m family hence one computationally efficient and surprisingly simple algorithm unifies old and new methods stability and accuracy analyses are presented for method orders m 2 3 and 4 to determine optimal parameters for implicit and explicit schemes along with numerical verification

adaptive finite element procedures for linear and non linear dynamics this paper discusses implementation and adaptivity of the discontinuous galerkin dg finite element method as applied to linear and non linear structural dynamic problems by the dg method both displacements and velocities are approximated as piecewise bilinear functions in space and time and may be discontinuous at the discrete time levels both implicit and explicit iterative algorithms for solving the resulted system of coupled equations are derived they are third order accurate and while the implicit procedure is unconditionally stable the explicit one is conditionally stable an h adaptive procedure based on the zienkiewicz zhu error estimate using the spr technique is applied numerical examples are presented to show the suitability of the dg method for both linear and non linear structural dynamic analysis copyright 1999 john wiley sons ltd

a methodology for the generation of low cost higher order methods for linear dynamics this work presents a methodology which generates efficient higher order methods for linear dynamics by improving the accuracy properties of norsett methods towards those of pade methods the methodology is based on a simple and low cost iterative procedure which is used to implement a set of higher order methods with controllable dissipation a sequence of improved solutions is obtained which correspond to algorithms offering an effective compromise between the efficiency of norsett methods and the accuracy of pade methods moreover a direct control over high frequency dissipation is possible by means of an algorithmic parameter numerical tests are reported which confirm that this set of algorithms is really attractive for linear dynamic analysis copyright 2003 john wiley sons ltd

multi field spacetime discontinuous galerkin methods for linearized elastodynamics we extend the single field spacetime discontinuous galerkin sdg method for linearized elastodynamics of abedi et al 1 to multi field versions a three field method in displacement velocity and strain is derived by invoking a bubnov galerkin weighted residuals procedure on the system of spacetime field equations and the corresponding jump conditions a two field formulation in displacement and velocity and the one field displacement formulation of 1 are obtained from the three field model through strong enforcement of kinematic compatibility relations all of these formulations balance linear and angular momentum at the element level and we prove that they are energy dissipative and unconditionally stable as in 1 we implement the sdg models using a causal advancing front meshing procedure that enables a patch by patch solution procedure with linear complexity in the number of spacetime elements numerical results show that the three field formulation is most efficient wherein all interpolated fields converge at the optimal o hp 1 rate for a given mesh size the three field model delivers error values that are more than an order of magnitude smaller than those of the one and two field models the three field formulation s efficiency is also superior independent of whether the comparison is based on matching polynomial orders or matching convergence rates

a unified set of single step algorithms part 1 general formulation and applications a general algorithm for a single step time marching scheme for use in dynamic or diffusion equations is presented this algorithm is easily programmed in its universal form for all orders of approximation and covers most of the currently used schemes as well as presenting many new possibilities in many cases it presents a computationally advantageous form over conventional procedures this is particularly so when compared with the newmark algorighm and its variants

crossed products byc0 x actions abstract suppose thatghas a representation grouph thatgab g g g is compactly generated and thatais ac algebra for which the complete regularization of prim a is a locally compact hausdorff spacex in a previous article we showed that there is a bijection z f between the collection of exterior equivalence classes of locally inner actions g aut a and the collection of principalgab bundlesz together with continuous functionsf x h2 g t in this paper we compute the crossed productsa gin terms of the dataz f andc h

a bott periodicity theorem for infinite dimensional euclidean space abstract we formulate and prove an equivariant bott periodicity theorem for infinite dimensional euclidean vector spaces the main features of our argument are i the construction of a non commutative c algebra to play the role of the algebra of functions on infinite dimensional euclidean space and ii the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the bott periodicity map these tools have applications to index theory and the novikov conjecture notably a proof of the novikov conjecture for amenable groups the applications will be considered elsewhere

solution of linear elastodynamics problems with space time finite elements on structured and unstructured meshes abstract a new implicit approach for the solution of linear elastodynamics problems on structured and unstructured meshes with space time finite elements is suggested new weak formulations are based on time continuous tcg and discontinuous tdg galerkin methods it was shown that for structured meshes the new tcg method has a higher order of accuracy by a factor of two than that of the standard tcg method at the same number of degrees of freedom the accuracy of the new tcg method is one order higher than the accuracy of the standard tdg method the introduction of new weighting functions for the weak formulations of the tcg and tdg methods allows control of the high frequency behavior without degradation of accuracy of the methods a variable number of elements in the time direction as well as a variable order of polynomial time approximations of finite elements for time slabs allows the implementation of a coupled space time adaptive procedure using structured or unstructured meshes numerical examples for 1 d problems demonstrate the effectiveness of the technique

finite element simulation of wave propagation in an axisymmetric bar an accurate solution for high frequency pulse propagation in an axisymmetric elastic bar is obtained using a new finite element technique that yields accurate non oscillatory solutions for wave propagation problems in solids the solution of the problem is very important for the understanding of dynamics experiments in the split hopkinson pressure bar shpb in contrast to known approaches no additional assumptions are necessary for the accurate solution of the considered problem the new solution helps to elucidate the complicated distribution of parameters during high frequency pulse propagation down the bar as well as to estimate the applicability of the traditional dispersion correction used in the literature for the analysis of wave propagation in a finite bar due to the dimensionless formulation of the problem the numerical results obtained depend on poisson s ratio the length of the bar and the pulse frequency and are independent of young s modulus the density and the radius of the bar

new linear codes constructed by concatenating extending and shortening methods corresp blokh and zyablov s concatenating method constructions x y1 of sloane et al and aiitop s construction xx are applied to various codes to obtain new binary linear codes these codes improve on the recent updated helgert stinaff table of verhoeff

the e theoretic descent functor for groupoids the paper establishes for a wide class of locally compact groupoids the e theoretic descent functor at the c algebra level in a way parallel to that established for locally compact groups by guentner higson and trout section 2 shows that actions on a c0 x algebra b where x is the unit space of can be usefully formulated in terms of an action on the associated bundle b section 3 shows that the functor b c b is continuous and exact and uses the disintegration theory of j renault section 4 establishes the existence of the descent functor under a very mild condition on the main technical difficulty involved being that of finding a algebra that plays the role of cb t b cont in the group case