Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{\text{ort}}^{0}\left( r,\,n \right)$ of stable rank $r$ orthogonal vector bundles on ${{\mathbb{P}}^{2}}$, with Chern classes $\left( {{c}_{1}},\,{{c}_{2}} \right)\,=\,\left( 0,\,n \right)$ and trivial splitting on the general line, is smooth irreducible of dimension $\left( r-2 \right)n\,-\,\left( _{2}^{r} \right)$ for $r\,=\,n$ and $n\,\ge \,4$, and $r\,=\,n-1$ and $n\,\ge \,8$. We speculate that the result holds in greater generality.

A category structure for Bratteli diagrams is proposed and a functor from the category of $\text{AF}$ algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli’s notion of equivalence, we obtain in particular a functorial formulation of Bratteli’s classification of $\text{AF}$ algebras (and at the same time, of Glimm’s classification of $\text{UHF}$ algebras). It is shown that the three approaches to classification of $\text{AF}$ algebras, namely, through Bratteli diagrams, $\text{K}$-theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.

The symmetric group ${{\mathcal{S}}_{n}}$ acts on the power set $\mathcal{P}\left( n \right)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

In this paper, we give a general explicit form of Cassels’ $p$-adic embedding theorem for number fields. We also give its refined form in the case of cyclotomic fields. As a byproduct, given an irreducible polynomial $f$ over $\mathbb{Z}$, we give a general unconditional upper bound for the smallest prime number $p$ such that $f$ has a simple root modulo $p$.

In this article we study exponential trichotomy for infinite dimensional discrete time dynamical systems. The goal of this article is to prove that finite time exponential trichotomy conditions allow us to derive exponential trichotomy for arbitrary times. We present an application to the case of pseudo orbits in some neighborhood of a normally hyperbolic set.

Let $\mathbf{TB}$ be the category of totally bounded, locally compact metric spaces with the ${{C}_{0}}$ coarse structures. We show that if $X$ and $Y$ are in $\mathbf{TB}$, then $X$ and $Y$ are coarsely equivalent if and only if their Higson coronas are homeomorphic. In fact, the Higson corona functor gives an equivalence of categories $\mathbf{TB}\,\to \,\mathbf{K}$, where $\mathbf{K}$ is the category of compact metrizable spaces. We use this fact to show that the continuously controlled coarse structure on a locally compact space $X$ induced by some metrizable compactification $\widetilde{X}$ is determined only by the topology of the remainder $\widetilde{X}\,\backslash \,X$.

We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and as symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.

We extend the classical notion of an outer action $\alpha $ of a group $G$ on a unital ring $A$ to the case when $\alpha $ is a partial action on ideals, all of which have local units. We show that if $\alpha $ is an outer partial action of an abelian group $G$, then its associated partial skew group ring $A\,{{\star }_{\alpha }}\,G$ is simple if and only if $A$ is $G$-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Let $w$ be either in the Muckenhoupt class of ${{A}_{2}}\left( {{\mathbb{R}}^{n}} \right)$ weights or in the class of $QC\left( {{\mathbb{R}}^{n}} \right)$ weights, and let ${{L}_{w}}\,:=\,-{{w}^{-1}}\,\text{div}\left( A\nabla \right)$ be the degenerate elliptic operator on the Euclidean space ${{\mathbb{R}}^{n}}$, $n\,\ge \,2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ associated with ${{L}_{w}}$ for $p\,\in \,(0,\,1]$, and when $p\,\in \,(\frac{n}{n+1},\,1]$ and $w\,\in \,{{A}_{{{q}_{0}}}}\left( {{\mathbb{R}}^{n}} \right)$ with ${{q}_{0}}\,\in \,[1,\,\frac{p(n+1)}{n})$, the authors prove that the associated Riesz transform $\nabla L_{w}^{-1/2}$ is bounded from $H_{{{L}_{w}}}^{p}\,\left( {{\mathbb{R}}^{n}} \right)$ to the weighted classical Hardy space $H_{w}^{p}\left( {{\mathbb{R}}^{n}} \right)$.