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Let $E \rightarrow M$ be a smooth vector bundle with a bilinear product on $\Gamma(E)$ satisfying the Jacobi identity. Assuming only the existence of an anchor map $\mathfrak{a}$ we show that $\mathfrak{a}([X,Y]) =…

We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle…

We discuss variants of Berikashvili's functor that arise in differential homological algebra, from simplicial bundles, from ordinary topological bundles, and in more general categorical settings. We prove that, under suitable circumstances,…

Let $\mathbb F$ be a finite field and let $\mathcal A$ and $\mathcal B$ be vector spaces of $\mathbb F$-valued continuous functions defined on locally compact spaces $X$ and $Y$, respectively. We look at the representation of linear…

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic…

We study threefolds X in a projective space having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise…

A Seifert manifold is a 3-dimensional manifold with a circle action. It is a circle bundle (with singularities) over a 2-dimensional orbifold. In this note, we discuss a generalized Seifert manifolds. By definition, they have bundle-like…

The main result of this paper is a proof of the continuity of a family of integral functionals defined on the space of functions of bounded variation with respect to a topology under which smooth functions are dense. These functionals occur…

A function from Baire space to the natural numbers is called formally continuous if it is induced by a morphism between the corresponding formal spaces. We compare formal continuity to two other notions of continuity on Baire space working…

For any Riemannian foliation F on a closed manifold M with an arbitrary bundle-like metric, leafwise heat flow of differential forms is proved to preserve smoothness on M at infinite time. This result and its proof have consequences about…

Let $W$ be a finite group and $T$ be an abelian group. Consider an extension $0 \ra T \ra N \ra W \ra 0$. For a smooth projective curve $X$, we give a precise description of the fiber of the quotient by $T$ map $q_T: \cM_X(N) \ra \cM_X(W)$…

We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing…

In this paper we provide an explicit connection between level-sets persistence and derived sheaf theory over the real line. In particular we construct a functor from 2-parameter persistence modules to sheaves over $\mathbb{R}$, as well as a…

We study Hurewicz fibrations between finite T$_0$--spaces from a combinatorial viewpoint and give strong conditions that a continuous map between finite T$_0$--spaces must satisfy in order to be a Hurewicz fibration. We also show that there…

We develop a theory of extensions of hyperfields that generalizes the notion of field extensions. Since hyperfields have a multivalued addition, we must consider two kinds of extensions that we call weak hyperfield extensions and strong…

In continuation of our work in Comm. in Algebra, vol. 28 (2000), we study ramified coverings of projective manifolds, in particular over Fano manifolds and investigate positivity properties of the associated vector bundle. Moreover we study…

In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector…

Let $V$ and $W$ be quiver representations over $\mathbb{F}_1$ and let $K$ be a field. The scalar extensions $V^K$ and $W^K$ are quiver representations over $K$ with a distinguished, very well-behaved basis. We construct a basis of…

We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.

The standard and fractional projections are extended from binary two-mode networks to weighted two-mode networks. Some interesting properties of the extended projections are proved.