We describe all complete and very ample linear series on 4-gonal curves.

Let $(R,{\frak m})$ be a commutative Noetherian complete local ring and let $I$ be a proper ideal of $R$ such that the category of all $I$-cofinite $R$-modules is an Abelian subcategory of the category of all $R$-modules. Then for each $I$-cofinite $R$-module $M$ it is shown that $\dim R/(I+{\rm Ann}_R M)=\dim M$. As a consequence of this result it is shown that if $J$ is an ideal of $R$ such that for each ${\frak p}\in {\rm mAss}_R R$, $\dim R/(J+{\frak p})\leq 1$ or ${\rm cd}(J,R/{\frak p})\leq 1$, then $\dim R/(J+{\rm Ann}_R M)=\dim M$, for each $J$-cofinite $R$-module $M$.

We consider coarse embeddings of real continuous functions spaces. We show that such an embedding can be well approximated by an affine surjective isometry under a certain condition of almost surjectivity.

We consider a class of Kirchhoff-Schrödinger-Poisson systems. By using the Mountain Pass Theorem and the Symmetric Mountain Pass Theorem of Rabinowitz, we prove the existence and multiplicity of nontrivial solutions. Our conditions weaken the Ambrosetti Rabinowitz type condition.

For $k\in\{2,3\}$ we classify endomorphisms of $H^*(G_{k,n};\mathbb Z_2)$ that commute with Steenrod squares (here, $G_{k,n}$ denotes the Grassmann manifold of $k$-dimensional subspaces in $\mathbb R^{n+k}$). Additionally, for all positive integers $k$ and $n$ we classify endomorphisms of $H^*(G_{k,n};\mathbb Z_2)$ that commute with Steenrod squares and such that the image of each class is a polynomial in the (nonzero) class of $H^1(G_{k,n};\mathbb Z_2)$.

Let $C(X)$ be the set of all real valued continuous functions on a metric space $(X,d)$. Caserta introduced the topology of strong Whitney convergence on bornology for $C(X)$, which is a generalization of the topology of strong uniform convergence on bornology introduced by Beer-Levi. The purpose of this paper is to study various cardinal invariants of the function space $C(X)$ endowed with the topologies of strong Whitney and Whitney convergence on bornology. In the process, we present simpler proofs of a number of results from the literature. In the end, relationships between cardinal invariants of strong Whitney convergence and strong uniform convergence on $C(X)$ have also been studied.

The Borel problem for Denjoy-Carleman and Braun-Meise-Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.

We introduce the concepts of complex polyanalytic weighted Bergman spaces and of quaternionic polyanalytic weighted Bergman spaces of first and second kind. In these spaces we then prove qualitative and quantitative results in approximation by polyanalytic polynomials. The quantitative approximation results are given in terms of higher order $L^{p}$-moduli of smoothness and in terms of the best approximation quantity.

We give a short proof of the fact that all subgroups of ${\rm PL}^+(I)$, the group of orientation preserving piecewise homeomorphisms of the unit interval, are inner amenable in the sense of Effros.