The Cauchy-Goursat Theorem for Rectifiable Jordan Curves

Rectifiable Measures, Square Functions Involving Densities, and the Cauchy Transform

This paper is devoted to the proof of two related results. The first one asserts that if μ is a Radon measure in R satisfying lim sup r→0 μ(B(x, r)) r > 0 and ∫ 1

The Jordan-Hölder Theorem

This submission contains theories that lead to a formalization of the proof of the Jordan-Hölder theorem about composition series of finite groups. The theories formalize the notions of isomorphism classes of groups, simple groups, normal series, composition series, maximal normal subgroups. Furthermore, they provide proofs of the second isomorphism theorem for groups, the characterization theo...

The Jordan-Hölder Theorem

The goal of this article is to formalize the Jordan-Hölder theorem in the context of group with operators as in the book [5]. Accordingly, the article introduces the structure of group with operators and reformulates some theorems on a group already present in the Mizar Mathematical Library. Next, the article formalizes the Zassenhaus butterfly lemma and the Schreier refinement theorem, and def...

Abnormal Curves on the Goursat Systems of Rn

The Cauchy-davenport Theorem for Semigroups

We generalize the Davenport transform to prove that, for A = (A,+) a cancellative unital semigroup and X,Y subsets of A such that the smallest subsemigroup generated by Y is commutative, one has that |X + Y | ≥ Ω(X,Y ) := min ( |X|+ |Y | − 1, sup y0∈Y× min y∈Y \{y0} ord(y − y0) ) if 2 ≤ |X|, |Y | < ∞. While extending the Cauchy-Davenport theorem to the broader and abstract setting of (possibly ...