Polynomial identities and noncommutative versal torsors

To any cleft Hopf Galois object, i.e., any algebra obtained from a Hopf algebra H by twisting its multiplication with a two-cocycle α, we attach two “universal algebras” and . The algebra is obtained by twisting the multiplication of H with the most general two-cocycle σ formally cohomologous to α. The cocycle σ takes values in the field of rational functions on H. By construction, is a cleft H-Galois extension of a “big” commutative algebra . Any “form” of can be obtained from by a specialization of and vice versa. If the algebra is simple, then is an Azumaya algebra with center . The algebra is constructed using a general theory of polynomial identities that we set up for arbitrary comodule algebras; it is the universal comodule algebra in which all comodule algebra identities of are satisfied. We construct an embedding of into ; this embedding maps the center of into when the algebra is simple. In this case, under an additional assumption, , thus turning into a central localization of . We completely work out these constructions in the case of the four-dimensional Sweedler algebra.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field

Let k be a real quadratic number field and the ring of integers and the group of units in k. Denote by the subgroup represented by elements of E of for a prime ideal in k. We show that for a given positive rational integer a, the set of prime numbers p for which the residual index of for lying above p is equal to a has a natural density c under the Generalized Riemann Hypothesis. Moreover, we give the explicit formula of c and conditions to c=0.     

Geometric degree of finite extensions of mappings from a smooth variety

Let f:V→W be a finite polynomial mapping of algebraic subsets V,W of and , respectively, with n≤m. It is known that f can be extended to a finite polynomial mapping . Moreover, it is known that, if V,W are smooth of dimension k,4k+2≤n=m, and f is dominated on every component (without vertical components) then there exists a finite polynomial extension such that , where means the number of points in the generic fiber of h. In this note we improve this result. Namely we show that there exists a finite polynomial extension such that .     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .     

Properties of cyclic subspace regression

Various properties of the regression vector produced by cyclic subspace regression with regard to the meancentered linear regression equation are put forth. In particular, the subspace associated with the creation of is shown to contain a basis that maximizes certain covariances with respect to , the orthogonal projection of onto a specific subspace of the range of X. This basis is constructed. Moreover, this paper shows how the maximum covariance values effect the . Several alternative representations of are also developed. These representations show that is a modified version of the l-factor principal components regression vector , with the modification occurring by a nonorthogonal projection. Additionally, these representations enable prediction properties associated with to be explicitly identified. Finally, methods for choosing factors are spelled out.     

Quadratic addition rules for quantum integers

For every positive integer n, the quantum integer [n]_q is the polynomial [n]_q=1+q+q²+?+q^n-1. A quadratic addition rule for quantum integers consists of sequences of polynomials , , and such that for all m and n. This paper gives a complete classification of quadratic addition rules, and also considers sequences of polynomials that satisfy the associated functional equation .     

Equivariant chain complexes, twisted homology and relative minimality of arrangements

We show that the π-equivariant chain complex (), , associated to a Morse-theoretic minimal CW-structure X on the complement of an arrangement , is independent of X. The same holds for all scalar extensions, , a field, where X is an arbitrary minimal CW-structure on a space M. When is a section of another arrangement , we show that the divisibility properties of the first Betti number of the Milnor fiber of  obstruct the homotopy realization of  as a subcomplex of a minimal structure on .If is aspherical and is a sufficiently generic section of , then may be described in terms of π, L and , for an arbitrary local system L; explicit computations may be done, when is fiber-type. In this case, explicit -presentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of , π_p(M), may also be obtained. For nonresonant abelian scalar extensions, the -rank of is combinatorially determined.     

A new majorization between functions, polynomials, and operator inequalities

Let P₊ be the set of all non-negative operator monotone functions defined on [0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a₁ and for t≧b₁. Then v₊?u₊ if m≦n and : in particular, for a sequence of orthonormal polynomials, (p_n-1)₊?(p_n)₊. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r).     

On the asymptotic number of inequivalent binary self-dual codes

Let Ψn be the number of inequivalent self-dual codes in . We prove that , where . Let Δn be the number of inequivalent doubly even self-dual codes in . We also prove that .     

Projective Schur groups of Henselian fields

One of the open questions that has emerged in the study of the projective Schur group of a field F is whether or not is an algebraic relative Brauer group over F, i.e. does there exist an algebraic extension L/F such that ? We show that the same question for the Schur group of a number field has a negative answer. For the projective Schur group, no counterexample is known. In this paper we prove that is an algebraic relative Brauer group for all Henselian valued fields F of equal characteristic whose residue field is a local or global field. For this, we first show how is determined by for an equicharacteristic Henselian field with arbitrary residue field k.     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .