In this paper we set up the foundations around the notions of formal differentiation and formal integration in the context of commutative Hopf algebroids and Lie–Rinehart algebras. Specifically, we construct a contravariant functor from the category of commutative Hopf algebroids with a fixed base algebra to that of Lie–Rinehart algebras over the same algebra, the differentiation functor, which can be seen as an algebraic counterpart to the differentiation process from Lie groupoids to Lie algebroids. The other way around, we provide two interrelated contravariant functors from the category of Lie–Rinehart algebras to that of commutative Hopf algebroids, the integration functors. One of them yields a contravariant adjunction together with the differentiation functor. Under mild conditions, essentially on the base algebra, the other integration functor only induces an adjunction at the level of Galois Hopf algebroids. By employing the differentiation functor, we also analyse the geometric separability of a given morphism of Hopf algebroids. Several examples and applications are presented.

We formulate, and provide strong evidence for, a natural generalization of a conjecture of Robert Coleman concerning Euler systems for the multiplicative group over arbitrary number fields.

The purpose of this paper is to contribute to the theory of profinite semigroups by considering the special class consisting of those all of whose finitely generated closed subsemigroups are countable, which are said to be locally countable. We also call locally countable a pseudovariety $\text{V}$ (of finite semigroups) for which all pro-$\text{V}$ semigroups are locally countable. We investigate operations preserving local countability of pseudovarieties and show that, in contrast with local finiteness, several natural operations do not preserve it. We also investigate the relationship of a finitely generated profinite semigroup being countable with every element being expressible in terms of the generators using multiplication and the idempotent (omega) power. The two properties turn out to be equivalent if there are only countably many group elements, gathered in finitely many regular $\mathcal{J}$-classes. We also show that the pseudovariety generated by all finite ordered monoids satisfying the inequality $1⩽x^n$ is locally countable if and only if $n=1$.

I review some problems on characters of finite solvable groups, while introducing new ones.

A new class of groups $\mathcal{C}$, containing all coherent RAAGs and all toral relatively hyperbolic groups, is defined. It is shown that, for a group $G$ in the class $\mathcal{C}$, the $\mathrm{\mathbb{Z}}\left[t\right]$-exponential group $G^{\mathrm{\mathbb{Z}}\left[t\right]}$ may be constructed as an iterated centraliser extension. Using this fact, it is proved that $G^{\mathrm{\mathbb{Z}}\left[t\right]}$ is fully residually $G$ (i.e. it has the same universal theory as $G$) and so its finitely generated subgroups are limit groups over $G$. If $\mathbb{G}$ is a coherent RAAG, then the converse also holds – any limit group over $\mathbb{G}$ embeds into ${\mathbb{G}}^{\mathrm{\mathbb{Z}}\left[t\right]}$. Moreover, it is proved that limit groups over $\mathbb{G}$ are finitely presented, coherent and $CAT\left(0\right)$, so in particular have solvable word and conjugacy problems.

In this paper we characterize the rank two vector bundles on ${\mathrm{\mathbb{P}}}^2$ which are invariant under the actions of the parabolic subgroups $G_p:={Stab}_p(PGL(3\left)\right)$ fixing a point in the projective plane, $G_L:={Stab}_L(PGL(3\left)\right)$ fixing a line, and when $p\in L$, the Borel subgroup $\mathbf{B}=G_p\cap G_L$ of $PGL\left(3\right)$. Moreover, we prove that the geometrical configuration of the jumping locus induced by the invariance does not, on the other hand, characterize the invariance itself. Indeed, we find infinite families that are almost uniform but not almost homogeneous.

In a previous paper, we constructed a smooth branch of travelling waves for the $2$-dimensional Gross–Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space. In particular, our result proves the nondegeneracy of these travelling waves, which is a key step in their classification and for the construction of multi-travelling waves.

Characterizing whether a Markov process of discrete random variables has a homogeneous continuous-time realization is a hard problem. In practice, this problem reduces to deciding when a given Markov matrix can be written as the exponential of some rate matrix (a Markov generator). This is an old question known in the literature as the embedding problem [11], which has been solved only for matrices of size $2\times 2$ or $3\times 3$. In this paper, we address this problem and related questions and obtain results along two different lines. First, for matrices of any size, we give a bound on the number of Markov generators in terms of the spectrum of the Markov matrix. Based on this, we establish a criterion for deciding whether a generic (distinct eigenvalues) Markov matrix is embeddable and propose an algorithm that lists all its Markov generators. Then, motivated and inspired by recent results on substitution models of DNA, we focus on the $4\times 4$ case and completely solve the embedding problem for any Markov matrix. The solution in this case is more concise as the embeddability is given in terms of a single condition.

Given a compact Sasaki manifold, we endow the space of the Sasaki potentials with an analogue of the Mabuchi metric. We show that its metric completion is negatively curved in the sense of Alexandrov.

Some crystalline measures supported by a Delone set $\mathrm{\Lambda }\subset {\mathrm{ℝ}}^2$ are constructed in this note. This gives a new proof of a remarkable theorem by Pavel Kurasov and Peter Sarnak.