global rigidity of higher rank abelian anosov algebraic actions we show that all c infty anosov mathbb z r actions on tori and nilmanifolds without rank one factor actions are up to c infty conjugacy actions by automorphisms

mixing automorphisms of compact groups and a theorem of schlickewei we prove that every mixing d by automorphisms of a compact connected abelian group is mixing of all orders

a new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method the purpose of this paper is to report on a new and efficient method for the evaluation of singular integrals in stress analysis of elastic and elasto plastic solids respectively by the direct boundary element method bem triangle polar co ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares the method was subsequently extended to the cubature of singular integrals over three dimensional internal cells as occur in applications of the bem to three dimensional elasto plasticity for this purpose so called tetrahedron polar co ordinates were introduced singular boundary integrals stretching over either linear triangular or quadratic quadilateral isoparametric boundry elements and singular volume integrals extending over either linear tetrahedral or quadratic hexahedral isoparametric internal cells are treated in case of higher order isoparametric boundary elements and internal cells division into a number of subelements and subcells respectively is necessary the analytical investigation is followed by a numerical study restricted to the use of quadratic quadrilateral isoparametric boundary elements this is justified by the fact that such elements as opposed to linear elements yield singular boundary integrals which cannot be integrated analytically the results of the numerical investigation demonstrate the potential of the developed concept

witt rings and brauer groups under multiquadratic extensions ii let f be a field of characteristic not 2 and suppose m is a multiquadratic extension of f that is m f is a finite abelian extension of exponent 2 so that m f for some finite subgroup g e f f where f 0 in studying brauer groups and products of quaternion algebras the second author was led to consider the homology groups n m f of a certain complex gm f associated to the extension m f this complex appears in 11 3 1 301 and in 1 1 below the purpose of the present work is to investigate the first homology group n m f and to exhibit its close connections with quadratic form theory the higher homology groups are examined in 11 301 the field f is said to have property pi n if n m f 1 for every multiquadratic extension m f with m f q 2 properties p l and p 2 always hold but there are examples 29 of fields f for which p 3 fails these examples are generalized in section 5

galois groups of maximal p extensions let p be an odd prime and f a field of characteristic different from p containing a primitive pth root of unity assume that the galois group g of the maximal p extension of f has a finite normal series with abelian factor groups then the commutator subgroup of g is abelian moreover g has a normal abelian subgroup with pro cyclic factor group if in addition f contains a primitive p 2 th root of unity then g has generators x y i i i with relations y i y j y j y i and xy i x 1 y i q 1 where q 0 or q p n for some n 1

a full tangent stiffness field boundary element formulation for geometric and material non linear problems of solid mechanics the field boundary element method naturally admits the solution algorithm in the incompressible regimes of fully developed plastic flow this is not the case with the generally popular finite element method without further modifications to the method such as reduced integration or a mixed method for treating the dilatational deformation the analyses by the field boundary element method for geometric and material non linear problems are generally carried out by an incremental algorithm where the velocities or displacement increments on the boundary are treated as the primary variables and an initial strain iteration method is commonly used to obtain the state of equilibrium for problems such as buckling and diffused tensile necking involving very large strains such a solution scheme may not be able to capture the bifurcation phenomena or the convergence will be unacceptably slow when the post bifurcation behaviour needs to be analysed to avoid this predicament a full tangent stiffness field boundary element formulation which takes the initial stress velocity gradient displacement gradient coupling terms accurately into account is presented in this paper here the velocity field both inside and on the boundary are treated as primary variables the large strain plasticity constitutive equation employed is based on an endochronic model of combined isotropic kinematic hardening plasticity using the concepts of material director triad and the associated plastic spin a generalized mid point radial return algorithm is presented for determining the objective increments of stress from the computed velocity gradients numerical results are presented for problems of diffuse necking involving very large strains and plastic instability in initially perfect elastic plastic plates under tension these results demonstrate the clear superiority of the full tangent stiffness algorithm over the initial strain algorithm in the context of the integral equation formulations for large strain plasticity

the quantitative behaviour of polynomial orbits on nilmanifolds a theorem of leibman asserts that a polynomial orbit g n g n z on a nilmanifold g g is always equidistributed in a union of closed sub nilmanifolds of g g in this paper we give a quantitative version of leibman s result describing the uniform distribution properties of a finite polynomial orbit g n g n n in a nilmanifold more specifically we show that there is a factorisation g eg where e n is smooth n g n z is periodic and rational and g n g n p is uniformly distributed up to a specified error d inside some subnilmanifold g g of g g for all sufficiently dense arithmetic progressions p n our bounds are uniform in n and are polynomial in the error tolerance d in a companion paper we shall use this theorem to establish the mobius and nilsequences conjecture from an earlier paper of ours

a new construction of noncrossed product algebras new examples of noncrossed product division algebras are obtained using methods different from all previous noncrossed product constructions the examples are division algebras over intersections of henselian valued fields and they have schur index and exponent p for any prime number p and any integers m n 2 n 3if 2 the basic tools used in the construction are valuation theory and galois cohomology no generic methods are applied and there is no p i theory along the way local global principles are proved for central simple algebras over intersections of p henselian valued fields the first examples of central simple division algebras which are not crossed products were obtained by amitsur in am in 1972 amitsur thereby settled a question that had been one of the outstanding open problems in the theory of algebras for at least thirty years his examples were the generic division algebras ud q n of index n over q the rational numbers for any natural number n such that p2 n p an odd prime or 8 all subsequent constructions of noncrossed products in ss 83 883 r r an ti have heretofore been based on amitsur s specialization argument and they are all generic division algebras or extensions of generic division algebras the centers of these noncrossed product algebras are not known nor are the brauer groups nor the absolute galois groups of the centers we present here a new method of constructing noncrossed product algebras in our approach the noncrossed product is realized as the underlying division algebra d of a tensor product of suitably chosen cyclic algebras over a field f l n l2 where each l is a pth root henselian valued field we prove local global principles relating the splitting fields of d to those of d f lt i 1 2 it is shown that the part of the brauer group of f is completely determined by that of ll and l2 computations for central simple f algebras thus become very tractable for exam ple we not only show that d is not a crossed product but also calculate exactly how large r must be so that the matrix ring mr d is a crossed product and how large s must be so that ms d is a tensor product of cyclic algebras indeed the structure of the brauer group of f is so nice that we were somewhat surprised that noncrossed products could possibly exist over f

a characterization of c fields via galois groups abstract it is known that those fields f which are of class c in the sense of r ware can be characterized by their witt ring structure as well as by the behavior of their binary quadratic forms and the structure of their galois 2 extensions in this paper we show that c fields which have at least 8 square classes are characterized by the absence of a certain group dc of order 16 as a galois group over them moreover we show that the level of such a field is determined by the presence or absence of the cyclic group of order 4 and the dihedral group of order 8 as galois groups over it these results provide an interesting continuation of the work of ware in the appendix we give an explicit construction of field extensions with galois group dc

direct evaluation of singular integrals in elastoplastic analysis by the boundary element method this paper presents a new method for the direct and accurate evaluation of strongly singular integrals in the sense of cauchy principal values and weakly singular integrals over quadratic boundary elements in three dimensional stress analysis and quadratic internal cells in two dimensional elastoplastic analysis by the boundary element method a quadratic triangle polar co ordinate transformation technique is applied to reduce the order of singularity of the singular integrals next a form of stokes theorem is introduced in order to remove the singularity in the cauchy principal value integrals therefore the evaluation of these integrals can be carried out by standard gaussian quadrature numerical examples of 2 d elastoplastic problems and a 3 d elastic problem show the effectiveness and efficiency of the method

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

on the quantitative distribution of polynomial nilsequences erratum this is an erratum to the paper the quantitative behaviour of polynomial orbits on nilmanifolds by the authors published as ann of math 2 175 2012 no 2 465 540 the proof of theorem 8 6 of that paper which claims a distribution result for multiparameter polynomial sequences on nilmanifolds was incorrect we provide two xes for this issue here first we deduce the equal sides case n1 nt n of this result from the 1 parameter results in the paper this is the same basic mode of argument we attempted originally though the details are dierent the equal sides case is the only one required in applications such as the proof of the inverse conjectures for the gowers norms due to the authors and ziegler to remove the equal sides condition one must rerun the entire argument of our paper in the context of multiparameter polynomial sequences g z t g rather than 1 parameter sequences g z g as is currently done a more detailed sketch of how this may be done is available online

the graded witt ring and galois cohomology ii a primary problem in the theory of quadratic forms over a field f of characteristic different from two is to prove that the rings hq f and gwf are isomorphic here hqf h gal fq f z 2z where fq is the quadratic closure of f and gwf is the graded witt nrng associated to the fundamental ideal of even dimensional forms in the witt ring wf of f in this paper we assume we are given a field extension k of f such that wk is close to wf or hq k is close to hq f a method is developed to obtain information about these graded rings over f and its 2 extensions from information about the corresponding graded ring of k this relative theory extends and includes the previously developed absolute case where k fq applications are also given to show that hq f and gwf are isomorphic for a collection of fields anrsing naturally from the theory of abstract witt rings

solvable absolute galois groups are metabelian we show that solvable absolute galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion free procyclic group in particular solvable absolute galois groups are metabelian moreover any field with solvable absolute galois group g admits a non trivial henselian valuation unless each sylow subgroup of g is either procyclic or isomorphic to z2 z 2z a complete classification of solvable absolute galois groups up to isomorphism is given

rigid elements valuations and realization of witt rings to the theory over the residue class field of the valuation this theorem generalizes to any 2 henselian valuation cf kl sect 12 21 let v be the valuation let a e k f o 1 be such that v a is not divisible by two the key idea behind the proof is that the quadratic form 1 a only represents elements in u a such an element a is called

powers of totally positive subgroups of fields abstract for a formally real field f a totally positive co torsion subgroup s of f and m 1 we prove that s m s m this extends a result of becker we prove several refinements of this fact related to higher pythagoras numbers real holomorphy rings and sums with mixed powers

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

demuskin fields with valuations we study the arithmetic structure of fields f of characteristic neq p containing a primitive pth root of unity for which the maximal pro p galois group of f is a finitely generated demuskin group

witt rings and almost free pro 2 groups let f be a field of characteristic not 2 wf the witt ring of quadratic forms over f if the maximal ideal consisting of even dimensional forms and i f its square let g 2 gal f 2 f where f 2 is the maximal 2 extension quadratic closure of f in this note we prove a decomposition theorem for wf in the case that z f is torsion free and use it to obtain a new proof of a theorem of ershov on the structure of almost free pro a groups e theorem 41 in what follows w f denotes the torsion subgroup of wf and wredf wf w f is the reduced witt ring the set of orderings on f will be denoted by x and will be given the harrison topology with subbasis consisting of the sets h a e x 1 0 a f is called a sap field if every open and closed clopen subset of x is equal to some h a if r r are abstract witt rings then r x r denotes their product in the category of witt rings ml all unexplained notations and terminology about quadratic forms and witt rings can be found in l and m for elementary facts about profinite groups the reader is referred to ri following kijima and nishi kn the field f is called quasi pythugorean if z f is torsion free i e u f d 2 equivalently if the kaplansky radical of f consists of all nonzero sums of two squares ell corollary 2 91 compare el2 classification theorem 23 examples of these fields include formally real generalized hilbert fields k pseudo real closed fields p and fields satisfying the strong hasse principle h4 every totally indefinite quadratic form of dimension 4 over f is isotropic i e ii f 2 a formally real field satisfies h if and only if it is a quasi pythagorean sap field elp in ko koziol gives a criterion characterizing the class of quasi pythagorean algebraic extensions of q such fields necessarily satisfy h the best known examples of fields satisfying h are formally real

a generalization of marshall s equivalence relation for p prime and for a field f containing a root of unity of order p we generalize marshall s equivalence relation on orderings to arbitrary subgroups of f of index p the equivalence classes then correspond to free pro p factors of the maximal pro p galois group of f we generalize to this setting results of jacob on the maximal pro 2 galois group of a pythagorean field

demuskin groups galois modules and the elementary type conjecture abstract let p be a prime and f p the maximal p extension of a field f containing a primitive pth root of unity we give a new characterization of demuskin groups among galois groups gal f p f when p 2 and assuming the elementary type conjecture when p 2 as well this characterization is in terms of the structure as galois modules of the galois cohomology of index p subgroups of gal f p f

design sensitivity coefficients for elasto viscoplastic problems by boundary element methods this paper is concerned with accurate and efficient determination of design sensitivity coefficients dscs for solid mechanics problems with small strains and rotations but with material non linearities present elasto plastic or elasto viscoplastic problems this approach is based on direct differentiation dda of the relevant derivative boundary element method dbem formulation of the problem analytical differentiation of the dbem equations leads to singular integral equations for the dscs with weakly logarithmically for 2 d singular kernels which are easy to deal with also stress components and their sensitivities are obtained on the boundary of a body with great accuracy these quantities are typically difficult to obtain accurately from finite element methods fem a computer program for general two dimensional plane strain and plane stress problems has been developed based on the above formulation numerical results are presented for some sample problems and these are compared against direct solutions the agreement between the dbem and direct solutions is excellent for these examples

a recursive description of pro p galois groups abstract in this note we study a modified version of the elementary type conjecture for pro p galois groups to be precise let g p f be the galois group of the maximal galois p extension of a field f containing a primitive p th root of unity under some natural assumptions concerning valuation rings of f and also orderings of f when p 2 we prove that g p f can be obtained from suitable closed subgroups using a finite number of free pro p products and semidirect products

global rigidity of higher rank anosov actions on tori and nilmanifolds an anosov diffeomorphism f on a torus t is affine if f lifts to an affine map on r by a classical result of franks and manning any anosov diffeomorphism g on t is topologically conjugate to an affine anosov diffeomorphism more precisely there is a homeomorphism t t such that f g 1 is an affine anosov diffeomorphism we call the franks manning conjugacy the linear part of f is the map induced by g on h1 t anosov diffeomorphisms are rarely c conjugate to affine ones for example one can perturb a linear anosov diffeomorphism locally around a fixed point p to change the conjugacy class of the derivative at p the resulting diffeomorphism will still be anosov but cannot be c conjugate to its linearization the situation is radically different for z actions with many anosov diffeomorphisms in other words anosov diffeomorphisms rarely commute with other anosov diffeomorphisms it follows easily from the result for a single anosov diffeomorphism that an anosov z action on t is topologically conjugate to a z action by affine anosov diffeomorphisms we call this action the linearization of and denote it by again for any a z the linear part of a is the map induced by a on h1 t the logarithms of the moduli of the eigenvalues of these linear parts define additive maps i z r which extend to linear functionals on r a weyl chamber of is a connected component of r i ker i theorem 1 1 let be a c action of z k 2 on a torus t and let be its linearization suppose that there is a z subgroup of z such that a is ergodic for every nonzero a z further assume that there is an anosov element for in each weyl chamber of then is c conjugate to furthermore for a linear z action on t having a z subgroup acting by ergodic elements is equivalent to several other properties in particular to being genuinely higher rank 37 a linear z action is called genuinely higher rank if for all finite index subgroups z of z no quotient of the z action factors through a finite extension of z hence we obtain the following corollary

regularity of functions smooth along foliations and elliptic regularity abstract we produce two sets of results arising in the analysis of the degree of smoothness of a function that is known to be smooth along the leaves of one or more foliations these foliations might arise from anosov systems and while each leaf is smooth the leaves might vary in a nonsmooth fashion one set of results gives microlocal regularity of such a function away from the conormal bundle of a foliation the other set of results gives local regularity of solutions to a class of elliptic systems with fairly rough coefficients such a regularity theory is motivated by one attack on the foliation regularity problem

advanced boundary element analysis of two and three dimensional problems of elasto plasticity an advanced formulation of the boundary element method has been developed for inelastic analysis based on an initial stress approach the iterative solution algorithm makes use of an accelerated initial stress approach in which the past history of initial stresses are used to obtain an initial estimate for the current increment in the present analysis the geometry and functions are represented by higher order quadratic shape functions to model complex geometries and rapid functional variations accurately the methods of numerical integration of the kernels particularly the singular type are substantially improved by devising suitable automatic sub segmentation routines that incorporate the recent developments in mapping procedures the formulations have been implemented for two dimensional plane stress plane strain and three dimensional elasto plasticity problems

mixing properties of commuting nilmanifold automorphisms we study mixing properties of commutative groups of automorphisms acting on compact nilmanifolds assuming that every non trivial element acts ergodically we prove that such actions are mixing of all orders we further show exponential 2 mixing and 3 mixing as an application we prove smooth cocycle rigidity for higher rank abelian groups of nilmanifold automorphisms

a new bem formulation for tow and three dimensional elastoplasticity using particular integras new two and three dimensional bem formulations are developed for elastoplastic stress analysis these new procedures differ from previous work in that volume integration is not required to incorporate the nonlinear effects in the analysis instead initial stress rates are introduced in the bem system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

a variable stiffness type boundary element formulation for axisymmetric elastoplastic media a new direct elastoplastic boundary element formulation for axisymmetric bodies is developed the procedure exploits certain features of the constitutive relationship to express the unknwon non linear initial stress rate tensor as a scalar quantity which can then be eliminated from the boundary equation system though a back substitution of the modified stress rate equations this formulation is implemented in a multi region system that utilizes quadratic isoparametric shape functions to model the geometry and field variables of the boundary and domain of a body the kernel functions are presented in appendix i and the singularities associated with these kernels are discussed new techniques are described to calculate the coefficients associated with the singular nodes

the kummerian property and maximal pro p galois groups abstract for a prime number p we give a new restriction on pro p groups g which are realizable as the maximal pro p galois group g f p for a field f containing a root of unity of order p this restriction arises from kummer theory and the structure of the maximal p radical extension of f we study it in the abstract context of pro p groups g with a continuous homomorphism g 1 p z p and characterize it cohomologically and in terms of 1 cocycles on g this is used to produce new examples of pro p groups which do not occur as maximal pro p galois groups of fields as above

linear equations in variables which lie in a multiplicative group let k be a field of characteristic 0 and let n be a natural number let r be a subgroup of the multiplicative group k n of finite rank r given a 1 a n e k write a a 1 a n for the number of solutions x x 1 x n of the equation a 1 x 1 a n x n 1 such that no proper subsum of a 1 x 1 a n x n vanishes we derive an explicit upper bound for a a 1 a n r which depends only on the dimension n and on the rank r

efficient evaluation of volume integrals in the boundary element method abstract nonlinear formulations of the boundary element method as for example elastoplastic analysis require integration over internal cells due to the singularity of the integrand this volume integration is cumbersome the paper presents a closed concept of direct evaluation of these domain integrals by means of simple transformations and standard gaussian quadrature rules for regular but nearly singular integrals an adaptive integration scheme is presented for the strongly singular case a regularized formulation based on gauss theorem is proposed numerical examples concerning elastoplastic analysis are included

computational aspects in 2d sbem analysis with domain inelastic actions the symmetric boundary element method applied to structures subjected to temperature and inelastic actions shows singular domain integrals in the present paper the strong singularity involved in the domain integrals of the stresses and tractions is removed and by means of a limiting operation this traction is evaluated on the boundary first the weakly singular domain integral in the somigliana identity s i of the displacements is regularized and the singular integral is transformed into a boundary one using the radial integration method subsequently using the differential operator applied to the displacement field the s i of the tractions inside the body is obtained and through a limit operation its expression is evaluated on the boundary the latter operation makes it possible to substitute the strongly singular domain integral in a strongly singular boundary one defined as a cauchy principal value with which the related free term is associated the expressions thus obtained for the displacements and the tractions in which domain integrals are substituted by boundary integrals were utilized in the galerkin approach for the evaluation in closed form of the load coefficients connected to domain inelastic actions this strategy makes it possible to evaluate the load coefficients avoiding considerable difficulties due to the geometry of the solid analyzed the obtained coefficients were implemented in the karnak sgbem calculus code copyright 2009 john wiley sons ltd

evaluation of the stress tensor in 3 d elastoplasticity by direct solving of hypersingular integrals a new method of direct numerical evaluation of hypersingular boundary integrals has been applied to the differentiated form of the somigliana identity hypersingular identity in 3d elastostatics through this method it is possible to evaluate the stress tensor on the boundary of a complex 3d structure in a very accurate manner by employing the direct boundary element method bem the geometry of the elements and their arrangements over the boundary of the structure are not subjected to any restrictions numerical examples illustrate the accuracy of the proposed method

rigidity of some abelian by cyclic solvable group actions on mathbb t n in this paper we study a natural class of groups that act as affine transformations of mathbb t n we investigate whether these solvable abelian by cyclic groups can act smoothly and nonaffinely on mathbb t n while remaining homotopic to the affine actions in the affine actions elliptic and hyperbolic dynamics coexist forcing a priori complicated dynamics in nonaffine perturbations we first show using the kam method that any small and sufficiently smooth perturbation of such an affine action can be conjugated smoothly to an affine action provided certain diophantine conditions on the action are met in dimension two under natural dynamical hypotheses we get a complete classification of such actions namely any such group action by c r diffeomorphisms can be conjugated to the affine action by c r varepsilon conjugacy next we show that in any dimension c 1 small perturbations can be conjugated to an affine action via c 1 varepsilon conjugacy the method is a generalization of the herman theory for circle diffeomorphisms to higher dimensions in the presence of a foliation structure provided by the hyperbolic dynamics

compatible valuations and generalized milnor k theory given a field and a subgroup of there is a minimal group for which there exists an compatible valuation whose units are contained in assuming that has finite index in and contains for prime we describe in computable theoretic terms

mixing of all orders of lie groups actions we show that for lie groups whose adjoint representation reflects their topology mixing implies mixing of all orders in particular we prove that mixing actions of a semisimple lie group are mixing of all orders answering a conjecture of b marcus