the graded witt group kernel of biquadratic extensions in characteristic two baeza showed that when char f 2 if e f is the separable biquadratic extension e f 1 b1 1 b2 then ker wq f wq e wf 1 b1 wf 1 b2 here we give the analogous result for the graded witt group specifically we obtain an exact sequence f n 1 f n 1 h2n 1f h2n 1e from which the result for gwqf follows by the isomorphisms of kato applications to 2 algebras of exponent and index 4 are also given

on the brauer group complex for a multiquadratic field extension abstract let f be a field of characteristic not 2 a 1 a n f m f a 1 a n let further g be the subgroup of f f 2 generated by a 1 a n for any i 1 n put a i i i a i a 1 consider the complex i k i k i 2 g f f 2 br 2 f res br 2 m n m e f e m m e 2 br 2 e where k i f a i are all the quadratic extensions of f containing in m w i i a i n k i f w i a i z a i z the homology group of this complex at i 1 th term from the left is denoted by h i m f independently of char f we give examples of these complexes with nontrivial h 3 m f and prescribed n 3 and a 1 a n 1 also similar examples concerning the group h 1 m f are constructed

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

quadratic form theory over preordered von neumann regular rings abstract to each pair r t consisting of a unitary commutative von neumann regular ring r where 2 is a unit and t is a preorder on r we associate a reduced special group g t r which faithfully reflects quadratic form theory modulo t over free r modules and then show using the representation of r as the ring of global sections of its affine scheme together with results from m dickmann f miraglia on quadratic forms whose total signature is zero mod 2 n solution to a problem of m marshall invent math 133 1998 243 278 m dickmann f miraglia lam s conjecture algebra colloq 10 2003 149 176 m dickmann f miraglia algebraic k theory of special groups j pure appl algebra 204 2006 195 234 that g t r satisfies a powerful k theoretic property the smc property from this we conclude that quadratic form theory modulo t over free r modules verifies marshall s signature conjecture lam s conjecture as well as a reduced version of milnor s witt ring conjecture

on the dec group of finite abelian galois extensions over global fields abstract if k f is a finite abelian galois extension of global fields whose galois group has exponent t we prove that there exists a short exact sequence 0 dec k f br t k f q p r q z z 0 where r q q and p is a finite set of primes of f that is empty if t is square free in particular we obtain that if t is square free then dec k f br t k f which we use to show that prime exponent division algebras over henselian valued fields with global residue fields are isomorphic to a tensor product of cyclic algebras finally we construct a counterexample to the result for higher exponent division algebras

on the first witt index of quadratic forms we prove hoffmann s conjecture determining the possible values of the first witt index of anisotropic quadratic forms of any given dimension the proof makes use of the steenrod type operations on the modulo 2 chow groups constructed by p brosnan

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

signal reconstruction from signed fourier transform magnitude in this paper we show that a one dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed fourier transform magnitude magnitude and 1 bit of phase information in addition we develop a numerical algorithm to reconstruct a one dimensional or multidimensional sequence from its fourier transform magnitude reconstruction examples obtained using this algorithm are also provided

on the generic splitting of quadratic forms in characteristic 2 in 9 and 10 knebusch established the basic facts of generic splitting theory of quadratic forms over a field of characteristic different from 2 this paper is related to 11 and 13 where knebusch and rehmann generalized partially this theory to a field of characteristic 2 more precisely we begin with a complete characterization of quadratic forms of height 1 we don t exclude anisotropic quadratic forms with quasi linear part of dimension at least 1 this allows us to extend the notion of degree to characteristic 2 we prove some results on excellent forms and splitting tower of a quadratic form some results on quadratic forms of height 2 and degree 1 or 2 are given

essential dimension of quadrics let x be an anisotropic projective quadric over a field f of characteristic not 2 the essential dimension dimes x of x a s def ined by oleg izhboldin is dimes x dim x i x 1 where i x is the first witt index of x i e the witt index of x over its function field let y be a complete possibly singular algebraic variety over f with all closed points of even degree and such that y has a closed point of odd degree over f x our main theorem states that dimes x dim y and that in the case dimes x dim y the quadric x is isotropic over f y applying the main theorem to a projective quadric y we get a proof of izhboldin s conjecture stated as follows if an anisotropic quadric y becomes isotropic over f x t hen dim es x dimes y and the equality holds if and only if x is isotropic over f y we also solve knebusch s problem by proving that the smallest transcendence degree of a generic splitting field of a quadric x is equal to dimes x let v be a non degenerate quadratic form of dimension at least 2 over a field f of characteristic not 2 and let x q be the quadric hypersurface given by the equation x 0 in the projective space p v we say that the quadric x is anisotropic if is an anisotropic quadratic form by springer s theorem every closed point of an anisotropic quadric x has

significance of group delay functions in signal reconstruction from spectral magnitude or phase in this paper we discuss the problem of signal reconstruction from spectral magnitude or phase using group delay functions we define two separate group delay functions for a signal one is derived from the magnitude and the other from the phase of the fourier transform of the signal the group delay functions offer insight into the problem of signal reconstruction and suggest methods for reconstructing signals from partial information such as spectral magnitude or phase we examine the problem of signal reconstruction from spectral magnitude or phase on the basis of these two group delay functions and derive the conditions for signal reconstruction based on existing iterative and noniterative algorithms for signal reconstruction we propose new algorithms for some special classes of signals the algorithms are illustrated with several examples our study shows that the relative importance of spectral magnitude and phase depends on the nature of signals speech signals are used to illustrate the importance of spectral magnitude and picture signals are used to illustrate the importance of phase in signal reconstruction problems using the group delay functions we explain the convergence behavior of the existing iterative algorithms for signal reconstruction

image representation by sign information image representation by sign data in its most general context is considered for the case when f is a two dimensional signal the authors study conditions under which f is determined by its sign or in other words by its quantization to 1 bit of information this study is carried out in two different directions first theoretical results are presented that set an algebraic condition under which real zero crossings uniquely specify a band limited image an interesting paradigm arising in theoretical computer vision is then posed are the zero crossing of f convolved with a laplacian of a gaussian at a single channel enough for unambiguously representing f second the problem of the completeness of the representation when the position of the zero crossings is known only approximately is addressed it is shown that when sign f is sampled significant ambiguities are introduced in the representation experimental results are presented which were obtained from an iterative algorithm devised to reconstruct real images from multiscale sign information

some remarks on units in grothendieck witt rings abstract we establish new structures on grothendieck witt rings including a g w k module structure on the unit group g w k and a presentation of g w as an infinite g m loop sheaf even though our constructions are motivated by speculations in stable a 1 homotopy theory our arguments are purely algebraic

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

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

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

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

semiinfinite cohomology of quantum groups we define a new cohomology theory of associative algebras called semiinfinite cohomology in the derived categories setting we investigate the case of a small quantum group u calculate semiinfinite cohomology spaces of the trivial u module and express them in terms of local cohomology of the nilpotent cone for the corresponding semisimple lie algebra we discuss the connection between the semiinfinite homology of u and the conformal blocks spaces

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

nonexcellence of multiquadratic field extensions abstract let k0 be a field chark0 2 n 2 a b 1 b n k 0 such that the elements a b 1 b n k 0 k 0 2 are linearly independent we show that there exists a field extension f k0 and an anisotropic 4 dimensional form over f such that d a the form f b 1 b n is isotropic and the form f b 1 b n an is not defined over f

stable signal and image reconstruction from noisy fourier transform phase the problem of reconstruction of finite signals and images from the noisy fourier transform phase is addressed the problem is pointed out to be ill posed with respect to distortion in the phase regularizing reconstruction algorithms are proposed that significantly improve estimates of signals or images reconstructed from their noisy phases

the behavior of quadratic and differential forms under function field extensions in characteristic two abstract let f be a field of characteristic 2 let n f be the f space of absolute differential forms over f there is a homomorphism n f n f d n 1 f given by x d x 1 x 1 d x n x n x 2 x d x 1 x 1 d x n x n mod d f n 1 let h n 1 f coker we study the behavior of hn 1 f under the function field f f where b1 bn is an n fold pfister form and f is the function field of the quadric 0 over f we show that ker h n 1 f h n 1 f f d b 1 b 1 d b n b n using kato s isomorphism of hn 1 f with the quotient inwq f in 1wq f where wq f is the witt group of quadratic forms over f and i w f is the maximal ideal of even dimensional bilinear forms over f we deduce from the above result the analogue in characteristic 2 of knebusch s degree conjecture i e inwq f is the set of all classes q with deg q n

quadratic forms and pfister neighbors in characteristic 2 we study pfister neighbors and their characterization over fields of characteristic 2 where we include the case of singular forms we give a somewhat simplified proof of a theorem of fitzgerald which provides a criterion for when a nonsingular quadratic form q is similar to a pfister form in terms of the hyperbolicity of this form over the function field of a form which is dominated by q from this we derive an analogue in characteristic 2 of a result by knebusch saying that in characteristic 2 a form is a pfister neighbor if its anisotropic part over its own function field is defined over the base field our result includes certain cases of singular forms but we also give examples which show that knebusch s result generally fails in characteristic 2 for singular forms as an application we characterize certain forms of height 1 in the sense of knebusch whose quasi linear parts are of small dimension we also develop some of the basics of a theory of totally singular quadratic forms this is used to give a new interpretation of the notion of the height of a standard splitting tower as introduced by the second author in an earlier paper

on representing signals using only timing information it is well known that only a special class of bandpass signals called real zero rz signals can be uniquely represented up to a scale factor by their zero crossings i e the time instants at which the signals change their sign however it is possible to invertibly map arbitrary bandpass signals into rz signals thereby implicitly represent the bandpass signal using the mapped rz signal s zero crossings this mapping is known as real zero conversion rzc in this paper a class of novel signal adaptive rzc algorithms is proposed specifically algorithms that are analogs of well known adaptive filtering methods to convert an arbitrary bandpass signal into other signals whose zero crossings contain sufficient information to represent the bandpass signal s phase and envelope are presented since the proposed zero crossings are not those of the original signal but only indirectly related to it they are called hidden or covert zero crossings cozecs the cozecs based representations are developed first for analytic signals and then extended to real valued signals finally the proposed algorithms are used to represent synthetic signals and speech signals processed through an analysis filter bank and it is shown that they can be reconstructed given the cozecs this signal representation has potential in many speech applications

powers of the fundamental ideal in the witt ring abstract let k be a field of characteristic different from 2 in the algebraic theory of quadratic forms one studies the witt ring w k of equivalence classes of non degenerate quadratic forms the witt ring has a filtration given by the powers in k of the fundamental ideal i k of even dimensional forms the ideal in k is generated by the set pn k of n fold pfister forms a1 an ni 1 1 ai where ai k k 0 many questions about this filtration and its quotients have arisen in the study of w k the spectacular work of voevodsky together with his collaborative work with orlov and vishik allows one to answer many old questions the purpose of this paper is to indicate some of these solutions

relations in in and inwq in characteristic 2 abstract let k be a field of characteristic 2 we give natural presentations of the subgroups i n k of the witt ring w k of k and the subgroups i n w q k of the witt group w q k of k our results generalize the results of arason and elman in j k arason r elman powers of the fundamental ideal in the witt ring j algebra 239 2001 150 160 where the characteristic is assumed to be different from 2

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

on the duality of time and frequency domain signal reconstruction from partial information certain results on one dimensional signal representation and reconstruction from partial information in the frequency domain are related to and derived using relevant results obtained in the time or positional information domain in particular we adopt logan s approach and apply his results concerning zero crossing representation of certain bandpass signals we also apply voelcker s results on demodulation of analytic ssb signals by an am detector exploiting the duality of the fourier stieltjes transform and its inverse we rederive and extend to continuous one dimensional signals some results concerning the representation of fourier transformed discrete time finite sequences by partial information such as one bit or complete fourier phase fourier magnitude or signed magnitude the duality and interrelationship of characteristics and constraints attributed to time and frequency signal representation create further possibilities of drawing results and interpretations from one to its counter domain in particular we indicate possible extensions of certain results to problems of n dimensional signals and to issues of stochastic signals we also highlight basic issues concerning the structure of signals and their reconstruction from partial information

the transfer ideal of quadratic forms and a hasse norm theorem mod squares any finite degree field extension k f determines an ideal jk f of the witt ring wf of f called the transfer ideal which is the image of any nonzero transfer map wk wf the ideal 59k f is computed for certain field extensions concentrating on the case where k has the form f va a ai e f when f and k are global fields we investigate whether there is a local global principle for membership in tkifthis is shown to be equivalent to the existence of a hasse norm theorem mod squares i e a local global principle for the image of the norm map nk f k k 2 f f 2 it is shown that such a hasse norm theorem holds whenever k f aj va although it does not always hold for more general extensions of global fields even some galois extensions with group z 2z x z 4z

differential forms and bilinear forms under field extensions abstract let f be a field of characteristic p 0 let n f be the f vector space of n differentials of f over f p let k f g be the function field of an irreducible polynomial g in m 1 variables over f we derive an explicit description of the kernel of the restriction map n f n k as an application in the case p 2 we determine the kernel of the restriction map when passing from the witt ring resp graded witt ring of symmetric bilinear forms over f to that over such a function field extension k

semi infinite induction and wakimoto modules the purpose of this paper is to suggest the construction and study properties of semi infinite induction which relates to semi infinite cohomology the same way induction relates to homology and coinduction to cohomology we prove a version of the shapiro lemma showing that the semi infinite cohomology of a module is isomorphic to that of the semi infinitely induced module a practical outcome of our construction is a simple construction of the wakimoto modules highest weight modules used in double sided bgg resolutions of irreducible modules

semi infinite cohomology and hecke algebras abstract this paper provides a homological algebraic foundation for generalizations of classical hecke algebras introduced in 1999 a sevostyanov comm math phys 204 137 these new hecke algebras are associated to triples of the form a a 0 e where a is an associative algebra over a field k containing subalgebra a 0 with augmentation e a 0 k these algebras are connected with cohomology of associative algebras in the sense that for every left a module v and right a module w the hecke algebra associated to triple a a 0 e naturally acts in the a 0 cohomology and a 0 homology spaces of v and w respectively we also introduce the semi infinite cohomology functor for associative algebras and define modifications of hecke algebras acting in semi infinite cohomology spaces we call these algebras semi infinite hecke algebras as an example we realize the w algebra w k g associated to a complex semisimple lie algebra g as a semi infinite hecke algebra using this realization we explicitly calculate the algebra w k g avoiding the bosonization technique used in 1992 b feigin and e frenkel internat j mod phys a 7 suppl 1a 197 215

bounds for the representation of quadratic forms abstract we prove first that for fixed integers n m 1 there is a uniform bound on the number of pfister forms of degree n over any pythagorean field f necessary to represent in the witt ring of f any form of dimension m as a linear combination of such forms with non zero coefficients in f uniform means that the bound does not depend either on the form or on the field f it is given by a recursive function f of n and m similar results hold for the reduced special groups arising from preordered fields and from fields whose pythagoras number is bounded by a fixed integer we single out a large class of pythagorean fields and more generally of reduced special groups cf 4 for which f has a simply exponential bound of the form cmn 1 c a constant such a class is closed under certain possibly infinitary operations which preserve marshall s signature conjecture in the case of groups of finite stability index s we obtain an upper bound for f which is quadratic on m 2n where c depends on s

decomposable and indecomposable algebras of degree 8 and exponent 2 we study the decomposition of central simple algebras of exponent 2 into tensor products of quaternion algebras we consider in particular decompositions in which one of the quaternion algebras contains a given quadratic extension let b be a biquaternion algebra over f a with trivial corestriction a degree 3 cohomological invariant is defined and we show that it determines whether b has a descent to f this invariant is used to give examples of indecomposable algebras of degree 8 and exponent 2 over a field of 2 cohomological dimension 3 and over a field m t where the u invariant of m is 8 and t is an indeterminate the construction of these indecomposable algebras uses chow group computations provided by merkurjev in appendix

annihilators of quadratic and bilinear forms over fields of characteristic two abstract let f be a field with 2 0 w f the witt ring of symmetric bilinear forms over f and w q f the w f module of quadratic forms over f let i f w f be the maximal ideal we compute explicitly in i f m and i m w q f the annihilators of n fold bilinear and quadratic pfister forms thereby answering positively in the case 2 0 certain conjectures stated by kruskemper in m kruskemper on annihilators in graded witt rings and in milnor s k theory in b jacob et al eds recent advances in real algebraic geometry and quadratic forms in contemp math vol 155 1994 pp 307 320

convergence of iterative nonexpansive signal reconstruction algorithms iterative algorithms for signal reconstruction from partial time and frequency domain knowledge have proven useful in a number of application areas in this paper a general convergence proof applicable to a general class of such iterative reconstruction algorithms is presented the proof relies on the concept of a nonexpansive mapping in both the time and frequency domains two examples studied in detail are time limited extrapolation equivalently band limited extrapolation and phase only signal reconstruction the proof of convergence for the phase only iteration is a new result obtained by this method of proof the generality of the approach allows the incorporation of nonlinear constraints such as time or space domain positivity or minimum and maximum value constraints finally the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique

isotropy of quadratic forms over the function field of a quadric in characteristic 2 abstract we extend to characteristic 2 a theorem by the first author which states that if and are anisotropic quadratic forms over a field f such that dim 2 n dim for some nonnegative integer n then stays anisotropic over the function field f of the case of singular forms is systematically included we give applications to the characterization of quadratic forms with maximal splitting we also prove a characteristic 2 version of a theorem by izhboldin on the isotropy of over f in the case dim 2 n 1 dim