Related papers: On Pseudo Algebraically Closed Extensions of Field…

In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…

We expand our previously founded basic theory of equiresidual algebraic geometry over an arbitrary commutative field, to a well-behaved theory of (equiresidual) algebraic varieties over a commutative field, thanks to the generalisation of…

Let K be an algebraically closed field of characteristic p. We exhibit a counterexample against a theorem asserted in one of our earlier papers, which claims to characterize the integral closure of K((t)) within the field of…

We present an extension-based approach for computing and verifying preferences in an abstract argumentation system. Although numerous argumentation semantics have been developed previously for identifying acceptable sets of arguments from…

Given a field $k$ and a finite group $H$, {\it{an $H$-parametric extension over $k$}} is a finite Galois extension of $k(T)$ of Galois group containing $H$ which is regular over $k$ and has all the Galois extensions of $k$ of group $H$…

Let $P$ be a Poisson algebra, $E$ a vector space and $\pi : E \to P$ an epimorphism of vector spaces with $V = {\rm Ker} (\pi)$. The global extension problem asks for the classification of all Poisson algebra structures that can be defined…

We give a detailed proof of Kolchin's results on differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. We closely follow former works due to Pillay and…

Reachability analysis evaluates system safety, by identifying the set of states a system may evolve within over a finite time horizon. In contrast to model-based reachability analysis, data-driven reachability analysis estimates reachable…

Probably Approximately Correct (i.e., PAC) learning is a core concept of sample complexity theory, and efficient PAC learnability is often seen as a natural counterpart to the class P in classical computational complexity. But while the…

Let $A$ be a unital associative algebra over a field $k$. All unital associative algebras containing $A$ as a subalgebra of a given codimension $\mathfrak{c}$ are described and classified. For a fixed vector space $V$ of dimension…

In the context of internal crossed modules over a fixed base object in a given semi-abelian category, we use the non-abelian tensor product in order to prove that an object is perfect (in an appropriate sense) if and only if it admits a…

Description logics (DLs) are well-known knowledge representation formalisms focused on the representation of terminological knowledge. Due to their first-order semantics, these languages (in their classical form) are not suitable for…

The inverse Galois problem asks whether any finite group can be realised as the Galois group of a Galois extension of the rationals. This problem and its refinements have stimulated a large amount of research in number theory and algebraic…

Abstract interpretation-based static analyses rely on abstract domains of program properties, such as intervals or congruences for integer variables. Galois connections (GCs) between posets provide the most widespread and useful formal tool…

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic…

The developments of Rademacher complexity and PAC-Bayesian theory have been largely independent. One exception is the PAC-Bayes theorem of Kakade, Sridharan, and Tewari (2008), which is established via Rademacher complexity theory by…

We apply the PAC-Bayes theory to the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-bounds) and explicit trade-off…

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives…

To generalize the notion of Galois closure for separable field extensions, we devise a notion of $G$-closure for algebras of commutative rings $R\to A$, where $A$ is locally free of rank $n$ as an $R$-module and $G$ is a subgroup of…

Let $G$ be an algebraic group, $X$ a generically free $G$-variety, and $K=k(X)^G$. A field extension $L$ of $K$ is called a splitting field of $X$ if the image of the class of $X$ under the natural map $H^1(K, G) \mapsto H^1(L, G)$ is…