Derivatives of Frobenius and Derivatives of Hodge–Tate Weights

In this paper we study the derivatives of Frobenius and the derivatives of Hodge–Tate weights for families of Galois representations with triangulations. We generalize the Fontaine–Mazur L-invariant and use it to build a formula which is a generalization of the Colmez–Greenberg–Stevens formula. For the purpose of proving this formula we show two auxiliary results called projection vanishing property and "projection vanishing implying L-invariants" property.     

Synthesis of Cyclopropenes and Fluorinated Cyclopropanes via Michael Initiated Ring Closure of Alkyl Triflones

A facile synthesis of cyclopropenes and fluorinated cyclopropanes from readily available alkyl triflones was developed. The reaction, regardless of electronic effect, gave products in good to excellent yields and moderate diastereoselectivity. The mechanism may involve tandem Michael addition of triflones/intramolecular nucleophilic cyclization (elimination of -SO₂CF₃)/elimination of fluoride.     

Finite-valued indistinguishability operators

Fuzzy equality relations or indistinguishability operators generalize the concepts of crisp equality and equivalence relations in fuzzy systems where inaccuracy and uncertainty is dealt with. They generate fuzzy granularity and are an essential tool in Computing with Words (CWW). Traditionally, the degree of similarity between two objects is a number between 0 and 1, but in many occasions this assignment cannot be done in such a precise way and the use of indistinguishability operators valued on a finite set of linguistic labels such as small, very much, etc. would be advisable. Recent advances in the study of finite-valued t-norms allow us to combine this kind of linguistic labels and makes the development of a theory of finite-valued indistinguishability operators and their application to real problems possible.     

Total synthesis of (±)-γ-lycorane via the electrocyclic ring closure of a divinylpyrroline

A concise total synthesis of (±)-γ-lycorane is described. The key step in the synthesis is the 6π-electrocyclic ring closure of a divinylpyrroline to give a tetrahydroindole, which is subsequently hydrogenated to give the all-cis indolizidine core.     

On a pairing for algebraic tori

Let T be an algebraic torus over a field F. There is a pairing between the groups of torsors for the torus T and its dual with values in the third Galois cohomology group over all field extensions of F. We study the kernel of this pairing.     

Cocycle deformations and Galois objects of semisimple Hopf algebras of dimension 16

Abstract

In this article, we determine cocycle deformations and Galois objects of non-commutative and non-cocommutative semisimple Hopf algebras of dimension 16. We show that these Hopf algebras are pairwise twist inequivalent mainly by calculating their higher Frobenius-Schur indicators, and that except three Hopf algebras which are cocycle deformations of dual group algebras, none of them admit non-trivial cocycle deformations.