Let $E$ be a stable rank 2 vector bundle on a smooth projective curve $X$ and $V\,\left( E \right)$ be the set of all rank 1 subbundles of $E$ with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, $E$, on $X$ with fixed $\deg \left( E \right)$ and $\deg \left( L \right),\,L\,\in \,V\left( E \right)$ and such that $\text{card}\,\left( V(E) \right)\,\ge \,2\,(\text{resp}\text{. card}\left( V(E) \right)\,=\,2)$.

We show that for $U$ and $V$ balanced open subsets of $\left( \text{Qno} \right)$ Fréchet spaces $E$ and $F$ that we have the topological identity

$$\text{(}\mathcal{H}(U\times V),{{\tau }_{\omega }})=\left( \mathcal{H}\left( U;\left( \mathcal{H}(V),{{\tau }_{\omega }} \right) \right),{{\tau }_{\omega }} \right).$$

Analogous results for the compact open topology have long been established. We also give an example to show that the $\left( \text{Qno} \right)$ hypothesis on both $E$ and $F$ is necessary.

In this paper, we show that the absolute value of the signature of the 2-parallel version of a link is less than or equal to the nullity of it and show that the signature, nullity, and Minkowski units of the 2-parallel version of a certain class of links are always equal to 0, 2, and 1 respectively.

We prove that Ornstein transformations are almost surely totally ergodic provided only that the cutting parameter is not bounded. We thus obtain a larger class of Ornstein transformations with the mixing property.

The purpose of this note is to give a simple combinatorial construction of the map from the canonically compactified moduli spaces of punctured complex projective lines to the moduli spaces ${{P}_{r}}$ of polygons with fixed side lengths in the Euclidean space ${{\mathbb{E}}^{3}}$. The advantage of this construction is that one can obtain a complete set of linear relations among the cycles that generate homology of ${{P}_{r}}$. We also classify moduli spaces of pentagons.

We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.

We present a new, simple proof of the classical Abel-Jacobi theorem using some elementary gauge theoretic arguments.

If $A$ is a prime ${{\text{C}}^{*}}$-algebra, $a\,\in \,A$ and $\lambda $ is in the numerical range $W\left( a \right)$ of $a$, then for each $\varepsilon \,>\,0$ there exists an element $h\,\in \,A$ such that $\left\| h \right\|\,=\,1$ and $\left\| {{h}^{*}}(a-\lambda )h \right\|\,<\,\varepsilon $. If $\lambda $ is an extreme point of $W\left( a \right)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general ${{\text{C}}^{*}}$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the ${{\text{C}}^{*}}$-convex hull of $a$ are characterized.

We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant.

As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.

The results herein continue observations on norm form equations and continued fractions begun and continued in the works [1]−[3], and [5]−[6].

Using spaces introduced by Anick, we construct a decomposition into indecomposable factors of the double loop spaces of odd primary Moore spaces when the powers of the primes are greater than the first power. If $n$ is greater than 1, this implies that the odd primary part of all the homotopy groups of the $2n\,+\,1$ dimensional sphere lifts to a mod ${{p}^{r}}$ Moore space.

We prove that ${{\left\{ \left( {{n}^{p}}-n \right)/P \right\}}_{p}}\in {{\Pi }_{p}}{{\text{F}}_{p}}$, with $p$ ranging over all primes, is independent of 1 over the integers, assuming a conjecture in elementary number theory generalizing the infinitude of Mersenne primes. This answers a question of Buium. We also prove a generalization.

For an integer $n$, let $d\left( n \right)$ denote the ordinary divisor function. This paper studies the asymptotic behavior of the sum

$$S\left( x \right)\,:=\sum\limits_{m\le x,n\le x}{d\left( {{m}^{2}}+{{n}^{2}} \right)}$$

It is proved in the paper that, as $x\,\to \,\infty $,

$$S(x):={{A}_{1}}{{x}^{2}}\log x+{{A}_{2}}{{x}^{2}}+{{O}_{\in }}({{x}^{\frac{3}{2}+\in }}),$$

where ${{A}_{1}}$ and ${{A}_{2}}$ are certain constants and $\in $ is any fixed positive real number.

The result corrects a false formula given in a paper of Gafurov concerning the same problem, and improves the error $O({{x}^{\frac{5}{3}}}\,{{(\log \,x)}^{9}})$ claimed by Gafurov.