quotients of absolute galois groups which determine the entire galois cohomology for a prime power q pd and a field f containing a root of unity of order q we show that the galois cohomology ring h g f mathbb z q is determined by a quotient g f 3 of the absolute galois group gf related to its descending q central sequence conversely we show that g f 3 is determined by the lower cohomology of gf this is used to give new examples of pro p groups which do not occur as absolute galois groups of fields

symbol length and stability index abstract we show that a pythagorean field more generally a reduced abstract witt ring has finite stability index if and only if it has finite 2 symbol length we give explicit bounds for the two invariants in terms of one another to approach the question whether those bounds are optimal we consider examples of pythagorean fields

numerical investigations for systems of second order periodic boundary value problems using reproducing kernel method the reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions and usually provides the solutions in term of rapidly convergent series in the appropriate hilbert spaces with components that can be elegantly computed the aim of the present analysis is to implement a relatively recent computational method reproducing kernel hilbert space for obtaining the solutions for systems of second order differential equations with periodic boundary conditions a reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied whilst the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points an efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory convergence analysis and error behavior of the presented method are also discussed in this approach computational results of some numerical examples are presented to illustrate the viability simplicity and applicability of the algorithm developed

compact embeddings and proper mappings abstract sufficient conditions are given to assert that a c 1 proper mapping defined on a compact embedding between banach spaces over k has a fixed point the proof of the result is based upon continuation methods

higher triple massey products and symbols abstract fix an arbitrary prime p let f be a field containing a primitive p th root of unity with absolute galois group g f and let h n denote its mod p cohomology group h n g f z p z the triple massey product of weight n k m n 3 is a partially defined multi valued function h n h k h m h n k m 1 in this work we prove that for an arbitrary prime p any defined 3mp of weight n 1 m where the first and third entries are symbols contains zero and that any defined 3mp of weight 1 k 1 where the middle entry is a symbol contains zero

witt groups of curves and surfaces we study witt groups of smooth curves and surfaces over algebraically closed fields of characteristic not two in both dimensions we determine both the classical witt group and balmer s shifted witt groups in the case of curves the results are supplemented with a complete description of the shifted grothendieck witt groups in a second step we analyse the relationship of witt groups of smooth complex curves and surfaces with their real topological k groups they turn out to be surprisingly close for all curves and for all projective surfaces of geometric genus zero the witt groups may be identified with the quotients of their even ko groups by the images of their complex topological k groups under realification

continuous embeddings and continuation methods abstract sufficient conditions are given to assert that a compact mapping defined on a continuous embedding between hilbert spaces over k has a fixed point the proof of the result is based upon continuation methods

projective modules over the real algebraic sphere of dimension 3 let a be a commutative noetherian ring of krull dimension 3 we give a necessary and sufficient condition for a projective modules of rank 2 to be free using this we show that all the finitely generated projective modules over the algebraic real 3 sphere are free c 2010 elsevier inc all rights reserved

existence and computation of zeros of perturbed mappings sufficient conditions are given to assert that a perturbed mapping has only one zero in a specific open ball this zero is estimated by means of a sequence of newton s iterations the proof of the result is constructive and is based upon continuation methods

birational invariants and 1 connectedness we study the a singular chain complex and zeroth a homology sheaf of smooth algebraic varieties over a field k these objects provide quadratic analogs of the motive and zeroth suslin homology sheaf of a smooth scheme in a sense we will explain we show that if k has characteristic 0 the zeroth a homology resp suslin homology sheaf is a birational invariant of smooth proper varieties both are a homotopy invariant we relate these sheaves to various types of unramified invariants in particular unramified cohomology we then show how the techniques in voevodsky s proof of the milnor conjecture and the voevodsky rost proof of its generalization the bloch kato conjecture can be used to study these sheaves using this last set of ideas we provide a construction of unirational non rational varieties where non rationality is detected by non triviality of a degree n unramified etale cohomology group with torsion coefficients yet cannot be detected by lower degree invariants

a cohomological classification of vector bundles on smooth affine threefolds we give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2 as a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties the proofs of these results involve three main ingredients first we give a description of the first nonstable a1 homotopy sheaf of the symplectic group second these computations can be used in concert with f morel s a1 homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques stemming from a version of the postnikov tower in a1 homotopy theory to reduce the classification results to cohomology vanishing statements third we prove the required vanishing statements

milnor witt k groups of local rings abstract we introduce milnor witt k groups of local rings and show that the n th milnor witt k group of a local ring r which contains an infinite field of characteristic not 2 is the pull back of the n th power of the fundamental ideal in the witt ring of r and the n th milnor k group of r over the n th milnor k group of r modulo 2 this generalizes the work of morel hopkins on milnor witt k groups of a field

the length and other invariants of a real field the length of a field is the smallest integer m such that any totally positive quadratic form of dimension m represents all sums of squares we investigate this field invariant and compare it to others such as the u invariant the pythagoras number the hasse number and the mordell function related to sums of squares of linear forms

the homogeneous spectrum of milnor witt k theory abstract for any field f of characteristic not equal to 2 we determine the zariski spectrum of homogeneous prime ideals in k m w f the milnor witt k theory ring of f as a corollary we recover lorenz and leicht s classical result on prime ideals in the witt ring of f our computation can be seen as a first step in balmer s program for studying the tensor triangular geometry of the stable motivic homotopy category

cohomological invariants for quadratic forms over local rings let a be local ring in which 2 is invertible and let n be a non negative integer we show that the nth cohomological invariant of quadratic forms is a well defined homomorphism from the nth power of the fundamental ideal in the witt ring of a to the degree n etale cohomology of a with mod 2 coefficients which is surjective and has kernel the hbox n 1 th power of the fundamental ideal this is obtained by proving the gersten conjecture for witt groups in an important mixed characteristic case

on the rationality problem for quadric bundles we classify all positive integers n and r such that stably nonrational complex r fold quadric bundles over rational n folds exist we show in particular that for any n and r a wide class of smooth r fold quadric bundles over pcn are not stably rational if r 2n 1 1 2n 2 in our proofs we introduce a generalization of the specialization method of voisin and of colliot thelene and pirutka which avoids universally ch0 trivial resolutions of singularities

a second zero of a function a sufficient condition is given to assert that a c m n 1 mapping between r m and r n which has a zero also has a second zero the condition can still be useful for continuous mappings when m n the constructive proof of the result is based upon continuation methods and supplies the existence of a path linking the zero points

on the vanishing of negative k groups we show that for a d dimensional scheme x essentially of finite type over an infinite perfect field k of characteristic p 0 the negative k groups k q x vanish for q d provided that strong resolution of singularities holds over the field k

on the u invariant of function fields of curves over complete discretely valued fields abstract let k be a complete discretely valued field with residue field if char k 0 char 2 and 2 d we prove that there exists an integer n depending on d such that the u invariant of any function field in one variable over k is bounded by n the method of proof is via introducing the notion of uniform boundedness for the p torsion of the brauer group of a field and relating the uniform boundedness of the 2 torsion of the brauer group to the finiteness of the u invariant we prove that the 2 torsion of the brauer group of function fields in one variable over k is uniformly bounded

unramified cohomology of quadrics i given a quadric x over a field f of characteristic 2 we compute the kernel and cokernel of the natural map in degree 4 from the mod 2 galois cohomology of f to the unramified mod 2 cohomology of f x when dim x 10 and in several smaller dimensional cases applications of these results to real quadrics and to the unramified witt ring are given

mappings sharing a value on finite dimensional spaces sufficient conditions are given to assert that differentiable mappings and differentiable proper mappings between r n and r m share a value the proof of the result is constructive and is based upon continuation methods

hyperbolicity and near hyperbolicity of quadratic forms over function fields of quadrics let p and q be anisotropic quadratic forms over a field f of characteristic ne 2 let s be the unique non negative integer such that 2 s mathrm dim p le 2 s 1 and let k denote the dimension of the anisotropic part of q after scalar extension to the function field f p of p we conjecture that the dimension of q must lie within k of an integer multiple of 2 s 1 this can be viewed as a direct generalization of hoffmann s separation theorem among other cases we prove that the conjecture is true if k 2 s 1 when k 0 this shows that any anisotropic form representing an element of the kernel of the natural restriction homomorphism w f rightarrow w f p has dimension divisible by 2 s 1

decomposability for division algebras of exponent two and associated forms this article investigates the structure of quadratic forms and of division algebras of exponent two over fields of characteristic different from two with the property that the third power of the fundamental ideal in the associated witt ring is torsion free

the homotopy limit problem for hermitian k theory equivariant motivic homotopy theory and motivic real cobordism abstract the homotopy limit problem for karoubi s hermitian k theory karoubi 1980 26 was posed by thomason 1983 44 there is a canonical map from algebraic hermitian k theory to the z 2 homotopy fixed points of algebraic k theory the problem asks roughly how close this map is to being an isomorphism specifically after completion at 2 in this paper we solve this problem completely for fields of characteristic 0 theorems 16 20 we show that the 2 completed map is an isomorphism for fields f of characteristic 0 which satisfy cd 2 f i but not in general

finiteness theorems for the shifted witt and higher grothendieck witt groups of arithmetic schemes abstract for smooth varieties over finite fields we prove that the shifted aka derived witt groups of surfaces are finite and the higher grothendieck witt groups aka hermitian k theory of curves are finitely generated for more general arithmetic schemes we give conditional results for example finite generation of the motivic cohomology groups implies finite generation of the grothendieck witt groups

euler class groups and the homology of elementary and special linear groups abstract we improve homology stability ranges for elementary and special linear groups over rings with many units our result implies stability for unstable quillen k groups and proves a conjecture of bass for commutative local rings with infinite residue fields we show that the obstruction to further stability is given by milnor witt k theory as an application we construct euler classes of projective modules with values in the cohomology of the milnor witt k theory sheaf for d dimensional commutative noetherian rings with infinite residue fields we show that the vanishing of the euler class is necessary and sufficient for an oriented projective module p of rank d to split off a rank 1 free direct summand along the way we obtain a new presentation of milnor witt k theory and of symplectic k 2 simplifying the classical matsumoto moore presentation

motivic spheres and the image of the suslin hurewicz map we show that an old conjecture of a a suslin characterizing the image of a hurewicz map from quillen k theory in degree n to milnor k theory in degree n admits an interpretation in terms of unstable mathbb a 1 homotopy sheaves of the general linear group using this identification we establish suslin s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3 we do this by linking the relevant unstable mathbb a 1 homotopy sheaf of the general linear group to the stable mathbb a 1 homotopy of motivic spheres

a compactness condition sufficient conditions are given to assert that a differentiable mapping between k m and k n where k r or k c has a zero the constructive proof of the result is based upon continuation methods and supplies the existence of a path leading to the zero point

the gersten conjecture for milnor k theory we prove that the n th milnor k group of an essentially smooth local ring over an infinite field coincides with the n n motivic cohomology of the ring this implies levine s generalized bloch kato conjecture

galois equivariance and stable motivic homotopy theory for a finite galois extension of fields l k with galois group g we study a functor from the g equivariant stable homotopy category to the stable motivic homotopy category over k induced by the classical galois correspondence we show that after completing at a prime and the motivic hopf map this results in a full and faithful embedding whenever k is real closed and l k i it is a full and faithful embedding after completion if a motivic version of serre s finiteness theorem is valid we produce strong necessary conditions on the field extension l k for this functor to be full and faithful along the way we produce several results on the stable c2 equivariant betti realization functor and prove convergence theorems for the p primary c2 equivariant adams spectral sequence

the pythagoras number and the u invariant of laurent series fields in several variables abstract we show that every sum of squares in the three variable laurent series field r x y z is a sum of 4 squares as was conjectured in a paper of choi dai lam and reznick in the 1980 s we obtain this result by proving that every sum of squares in a finite extension of r x y is a sum of 3 squares it was already shown in choi dai lam and reznick s paper that every sum of squares in r x y itself is a sum of two squares we give a generalization of this result where r is replaced by an arbitrary real field our methods yield similar results about the u invariant of fields of the same type

dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl in this paper we study the dynamic stability of the three dimensional axisymmetric navier stokes equations with swirl to this purpose we propose a new one dimensional model that approximates the navier stokes equations along the symmetry axis an important property of this one dimensional model is that one can construct from its solutions a family of exact solutions of the three dimensionafinal navier stokes equations the nonlinear structure of the one dimensional model has some very interesting properties on one hand it can lead to tremendous dynamic growth of the solution within a short time on the other hand it has a surprising dynamic depletion mechanism that prevents the solution from blowing up in finite time by exploiting this special nonlinear structure we prove the global regularity of the three dimensional navier stokes equations for a family of initial data whose solutions can lead to large dynamic growth but yet have global smooth solutions