We consider when a single submodule and also when every submodule of a module $M$ over a general ring $R$ has a unique closure with respect to a hereditary torsion theory on Mod-$R$.

In this paper, we prove that the unboundedness of ranks in families of Jacobian varieties of twisted Fermat curves is equivalent to the divergence of certain infinite series.

In this paper, we explore a generalization of the notion of integrality. In particular, we study a near-integrality condition that is intermediate between the concepts of integral and almost integral. This property (referred to as the $\Omega $-almost integral property) is a representative independent specialization of the standard notion of almost integrality. Some of the properties of this generalization are explored in this paper, and these properties are compared with the notion of pseudo-integrality introduced by Anderson, Houston, and Zafrullah. Additionally, it is shown that the $\Omega $-almost integral property serves to characterize the survival/lying over pairs of Dobbs and Coykendall.

We prove that the $\mathfrak{S}$-module PreLie is a free Lie algebra in the category of $\mathfrak{S}$-modules and can therefore be written as the composition of the $\mathfrak{S}$-module Lie with a new $\mathfrak{S}$-module $X$. This implies that free pre-Lie algebras in the category of vector spaces, when considered as Lie algebras, are free on generators that can be described using $X$. Furthermore, we define a natural filtration on the $\mathfrak{S}$-module $X$. We also obtain a relationship between $X$ and the $\mathfrak{S}$-module coming from the anticyclic structure of the PreLie operad.

We show that for every separable Banach space $X$, either $\text{S}{{\text{P}}_{w}}\left( X \right)$ (the set of all spreading models of $X$ generated by weakly-null sequences in $X$, modulo equivalence) is countable, or $\text{S}{{\text{P}}_{w}}\left( X \right)$ contains an antichain of the size of the continuum. This answers a question of S. J. Dilworth, E. Odell, and B. Sari.

We characterize maps between $n$-dimensional Nöbeling manifolds that can be approximated by homeomorphisms.

In this paper, we give two characterizations of the exactness of operator spaces.

It is shown that an old direct argument of Erdős and Heilbronn may be elaborated to yield a result of the current inverse type.

An integral domain $R$ is IDPF (Irreducible Divisors of Powers Finite) if, for every non-zero element $a$ in $R$, the ascending chain of non-associate irreducible divisors in $R$ of ${{a}^{n}}$ stabilizes on a finite set as $n$ ranges over the positive integers, while $R$ is atomic if every non-zero element that is not a unit is a product of a finite number of irreducible elements (atoms). A ring extension $S$ of $R$ is a root extension or radical extension if for each $s$ in $S$, there exists a natural number $n\left( s \right)$ with ${{s}^{n\left( s \right)}}$ in $R$. In this paper it is shown that the ascent and descent of the IDPF property and atomicity for the pair of integral domains $\left( R,\,S \right)$ is governed by the relative sizes of the unit groups $\text{U}\left( R \right)$ and $\text{U}\left( S \right)$ and whether $S$ is a root extension of $R$. The following results are deduced from these considerations: An atomic IDPF domain containing a field of characteristic zero is completely integrally closed. An affine domain over a field of characteristic zero is IDPF if and only if it is completely integrally closed. Let $R$ be a Noetherian domain with integral closure $S$. Suppose the conductor of $S$ into $R$ is non-zero. Then $R$ is IDPF if and only if $S$ is a root extension of $R$ and $\text{U}\left( S \right)/\text{U}\left( R \right)$ is finite.

We give an infinite family of congruent number elliptic curves each with rank at least three.

An example is given of a UFD which has an infinitely generated Derksen invariant. The ring is “almost rigid” meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's ABC-theorem.

Let $\Gamma $ be a discrete group and let $f\,\in \,{{l}^{1}}(\Gamma )$. We observe that if the natural convolution operator $\rho f\,:\,{{l}^{\infty }}(\Gamma )\,\to \,{{l}^{\infty }}(\Gamma )$ is injective, then $f$ is invertible in ${{l}^{1}}(\Gamma )$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra ${{l}^{1}}(\Gamma )$.

We give simple examples to show that in general one cannot replace ${{l}^{\infty }}$ with ${{l}^{p}},\,1\,\le \,p\,<\,\infty $, nor with ${{L}^{\infty }}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma $, and give some partial results.

We prove the equivalence of three definitions given by different comparison relations for infiniteness of positive elements in simple ${{C}^{*}}$-algebras.

Let $R$ be a commutative Noetherian ring and $\mathfrak{a}$ a proper ideal of $R$. We show that if $n\,:=\,\text{grad}{{\text{e}}_{R}}\,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))\,\cong \,\text{Ext}_{R}^{n}(H_{\mathfrak{a}}^{n}(R),\,R)$. We also prove that, for a nonnegative integer $n$ such that $H_{\mathfrak{a}}^{i}(R)\,=\,0$ for every $i\,\ne \,n$, if $\text{Ext}_{R}^{i}({{R}_{z}},\,R)\,=\,0$ for all $i\,>\,0$ and $z\,\in \,\mathfrak{a}$, then $\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$ is a homomorphic image of $R$, where ${{R}_{z}}$ is the ring of fractions of $R$ with respect to a multiplicatively closed subset $\{{{z}^{j}}|j\,\ge \text{0}\}$ of $R$. Moreover, if $\text{Ho}{{\text{m}}_{R}}({{R}_{z}},R)=0$ for all $z\,\in \,\mathfrak{a}$, then ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is an isomorphism, where ${{\mu }_{H_{\mathfrak{a}}^{n}(R)}}$ is the canonical ring homomorphism $R\,\to \,\text{En}{{\text{d}}_{R}}(H_{\mathfrak{a}}^{n}(R))$.

We prove that the group of rational points of a non-isotrivial elliptic curve defined over the perfect closure of a function field in positive characteristic is finitely generated.

We present some partial results concerning two conjectures of Schützenberger on evacuations of Young tableaux.

Let $\phi$ be a Drinfeld module of generic characteristic, and let $X$ be a sufficiently generic affine subvariety of $\mathbb{G}_{a}^{g}$. We show that the intersection of $X$ with a finite rank $\phi$-submodule of $\mathbb{G}_{a}^{g}$ is finite.

In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.

We study a semilinear elliptic problem on a compact Riemannian manifold with boundary, subject to an inhomogeneous Neumann boundary condition. Under various hypotheses on the nonlinear terms, depending on their behaviour in the origin and infinity, we prove multiplicity of solutions by using variational arguments.

We give weighted norm inequalities for the maximal fractional operator ${{\mathcal{M}}_{q}},\beta $ of Hardy–Littlewood and the fractional integral ${{I}_{\gamma }}$. These inequalities are established between ${{\left( {{L}^{q}},\,{{L}^{p}} \right)}^{\alpha }}\left( X,\,d,\,\mu \right)$ spaces (which are superspaces of Lebesgue spaces ${{L}^{\alpha }}\left( X,\,d,\,\mu \right)$ and subspaces of amalgams $\left( {{L}^{q}},\,{{L}^{p}} \right)\left( X,d,\mu \right)$) and in the setting of space of homogeneous type $\left( X,d,\mu \right)$. The conditions on the weights are stated in terms of Orlicz norm.