We settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, i.e., to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra.

A field $F$ is said to be tractable when a condition described below on the simultaneous representation of quaternion algebras holds over $F$ . It is shown that a global field $F$ is tractable iff $F$ has at most one dyadic place. Several other examples of tractable and nontractable fields are given.

We develop a method of separable reduction for Fréchet-like normals and $\epsilon$ -normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fréchet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$ -normals.

Soit $k$ un corps de caractéristique $p$ quelconque. Nous définissons la catégorie des $k$ -algèbres de cochaînes fortement quasi-commutatives et nous donnons une condition nécessaire et suffisante pour que l’algèbre de cohomologie à coefficients dans ${{\text{Z}}_{2}}$ d’un objet de cette catégorie soit un module instable sur l’algèbre de Steenrod à coefficients dans ${{\text{Z}}_{2}}$ .

A tout c.w. complexe simplement connexe de type fini $X$ on associe une $k$ -algèbre de cochaînes fortement quasi-commutative; la structure de module sur l’algèbre de Steenrod définie sur l’algèbre de cohomologie de celle-ci coïncide avec celle de ${{H}^{*}}(X;\,{{\text{Z}}_{2}})$ .

A classical theorem of Hermite and Joubert asserts that any field extension of degree $n\,=\,5\,\text{or}\,\text{6}$ is generated by an element whose minimal polynomial is of the form ${{\lambda }^{n}}\,+\,{{c}_{1}}{{\lambda }^{n-1}}\,+\,\cdot \cdot \cdot +\,{{c}_{n-1}}\lambda \,+\,{{c}_{n}}$ with ${{c}_{1\,}}\,=\,\,{{c}_{3}}\,=\,0$ . We show that this theorem fails for $n\,=\,{{3}^{m}}$ or ${{3}^{m}}+{{3}^{l}}$ (and more generally, for $n={{p}^{m}}$ or ${{p}^{m}}+{{p}^{l}}$ , if 3 is replaced by another prime $p$ ), where $m\,>\,1\,\ge \,0$ . We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$ .

We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$ .

If $G$ is a closed subgroup of a commutative hypergroup $K$ , then the coset space $K/G$ carries a quotient hypergroup structure. In this paper, we study related convolution structures on $K/G$ coming fromdeformations of the quotient hypergroup structure by certain functions on $K$ which we call partial characters with respect to $G$ . They are usually not probability-preserving, but lead to so-called signed hypergroups on $K/G$ . A first example is provided by the Laguerre convolution on $[0,\infty [$ , which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair $\left( U\left( n,1 \right),\,U\left( n \right) \right)$ are discussed.

We construct meromorphic functions with asymptotic power series expansion in ${{z}^{-1}}$ at $\infty$ on an Arakelyan set $A$ having prescribed zeros and poles outside $A$ . We use our results to prove approximation theorems where the approximating function fulfills interpolation restrictions outside the set of approximation.

We study the correspondence of representations arising by restricting the minimal representation of the linear group of type ${{E}_{7}}$ and relative rank 4. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of ${{G}_{2}}$ and $\text{P}{{\text{U}}_{3}}\left( D \right)$ .

Let $m$ be a point of the maximal ideal space of ${{H}^{\infty }}$ with nontrivial Gleason part $P\left( m \right)$ . If ${{L}_{m}}\,:\,\text{D}\,\to \,\text{P(m)}$ is the Hoffman map, we show that ${{H}^{\infty }}\,\circ \,{{L}_{m}}$ is a closed subalgebra of ${{H}^{\infty }}$ . We characterize the points $m$ for which ${{L}_{m}}$ is a homeomorphism in terms of interpolating sequences, and we show that in this case ${{H}^{\infty }}\,\circ \,{{L}_{m}}$ coincides with ${{H}^{\infty }}$ . Also, if ${{I}_{m}}$ is the ideal of functions in ${{H}^{\infty }}$ that identically vanish on $P\left( m \right)$ , we estimate the distance of any $f\,\in \,{{H}^{\infty }}\,\text{to}\,{{I}_{m}}$ .

Let $U\left( n,\,n \right)$ be the rank $n$ quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of $U\left( n,\,n \right)$ has at most simple poles at the integers or half integers in certain strip of the complex plane.

Let $d$ be the discriminant of an imaginary quadratic field. Let $a$ , $b$ , $c$ be integers such that

$${{b}^{2}}-4ac=d,a>0,\gcd (a,b,c)=1.$$

The value of $\left| \eta ((b+\sqrt{d})/\left. 2a) \right| \right.$ is determined explicitly, where $\eta \left( z \right)$ is Dedekind’s eta function

$$\eta (z)\,=\,{{e}^{\pi iz/12}}\,\prod\limits_{m=1}^{\infty }{(1-{{e}^{2\pi imz}})\,\,\,(im(z)>0).}$$