Let $F$ be a totally real number field and let $\text{G}{{\text{L}}_{n}}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and ${{F}_{\mathfrak{p}}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\text{G}{{\text{L}}_{n}}$ over the field ${{F}_{\mathfrak{p}}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

We find a lower bound on the number of imaginary quadratic extensions of the function field ${{\mathbb{F}}_{q}}\left( T \right)$ whose class groups have an element of a fixed order.

More precisely, let $q\,\ge \,5$ be a power of an odd prime and let $g$ be a fixed positive integer $\ge \,3$. There are $\gg \,{{q}^{\ell \left( \frac{1}{2}+\frac{1}{g} \right)}}$ polynomials $D\,\in \,{{\mathbb{F}}_{q}}\left[ T \right]$ with $\deg \left( D \right)\,\le \,\ell $ such that the class groups of the quadratic extensions ${{\mathbb{F}}_{q}}\left( T,\,\sqrt{D} \right)$ have an element of order $g$.

We classify finite subgroups of $\text{SO}\left( 4 \right)$ generated by anti-unitary involutions. They correspond to involutions fixing pointwise a Lagrangian plane. Explicit descriptions of the finite groups and the configurations of Lagrangian planes are obtained.

Functions defined on closed sets are simultaneously approximated and interpolated by meromorphic functions with prescribed poles and zeros outside the set of approximation.

In answer to a question posed in [3], we give sufficient conditions on a Lie nilmanifold so that any ergodic rotation of the nilmanifold is metrically conjugate to its inverse. The condition is that the Lie algebra be what we call quasi-graded, and is weaker than the property of being graded. Furthermore, the conjugating map can be chosen to be an involution. It is shown that for a special class of groups, the condition of quasi-graded is also necessary. In certain examples there is a continuum of conjugacies.

In this paper we find a formula for the Alexander polynomial ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$ of pretzel knots and links with $\left( {{p}_{1}},...,{{p}_{k}},-1 \right)$ twists, where $k$ is odd and ${{p}_{1}},...,{{p}_{k}}$ are positive integers. The polynomial ${{\Delta }_{2,3,7}}\left( x \right)$ is the well-known Lehmer polynomial, which is conjectured to have the smallest Mahler measure among all monic integer polynomials. We confirm that ${{\Delta }_{2,3,7}}\left( x \right)$ has the smallest Mahler measure among the polynomials arising as ${{\Delta }_{p1,...,{{p}_{k}}\left( x \right)}}$.

Let $\mathcal{E}$ be an ample vector bundle of rank $r$ on a projective variety $X$ with only log-terminal singularities. We consider the nefness of adjoint divisors ${{K}_{X}}\,+\,\left( t-r \right)\,\det \,\mathcal{E}$ when $t\,\ge \,\dim\,X$ and $t\,>\,r$. As an application, we classify pairs $\left( X,\,\mathcal{E} \right)$ with ${{c}_{r}}$-sectional genus zero.

Nous montrons que la $\text{A}$-catégorie d’un espace simplement connexe de type fini est inférieure ou égale à $n$ si et seulement si son modèle d’Adams-Hilton est un rétracte homotopique d’une algèbre différentielle à $n$ étages. Nous en déduisons que l’invariant Acat augmente au plus de 1 lors de l’attachement d’une cellule à un espace.

In this article it is shown that every bounded linear operator on a complex, infinite dimensional, separable Hilbert space is a sum of at most eighteen unilateral (alternatively, bilateral) weighted shifts. As well, we classify products of weighted shifts, as well as sums and limits of the resulting operators.

We prove an identity between weighted orbital integrals of the unit elements in the Hecke algebras of $\text{GL}\left( r \right)$ and its $n$-fold metaplectic covering, under the assumption that $n$ is relatively prime to any proper divisor of every $1\,\le \,j\,\le \,r$.

We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.

We show that, if a Banach algebra $\mathfrak{A}$ is a left ideal in its second dual algebra and has a left bounded approximate identity, then the weak amenability of $\mathfrak{A}$ implies the $\left( 2m+1 \right)$-weak amenability of $\mathfrak{A}$ for all $m\,\ge \,1$.