In the field of Arabic sociolinguistics, diglossia has been an interesting linguistic inquiry since it was first discussed by Ferguson in 1959. Since then, diglossia has been discussed, expanded, and revisited by Badawi (1973), Hudson (2002), and Albirini (2016) among others. While the discussion of the Arabic diglossic situation highlights the existence of two separate codes (High and Lo...

Introduction 2 1. Normed Rings 4 1.1. Elementwise Norms 4 1.2. Ideal norms 5 1.3. Finite Quotient Domains 5 1.4. Euclidean norms 6 2. Euclidean quadratic forms and ADC forms 7 2.1. Euclidean quadratic forms 7 2.2. Euclideanity 7 2.3. ADC-forms 8 2.4. Euclidean Forms are ADC Forms 8 2.5. The Generalized Cassels-Pfister Theorem 9 2.6. Maximal Lattices 10 3. Localization and Completion 11 3.1. Loc...

We have three motivations for looking at this theorem. Firstly it can be seen as partial confirmation of a conjecture of Fontaine and Mazur which asserts that if ρ : GQ → GLn(Qp) is a continuous irreducible representation ramified at only finitely many primes and such that the image of the inertia group at p is finite, then the image of ρ is finite (see [FM]). Our theorem verifies this conjectu...

We present an account of negative differential forms within a natural algebraic framework of differential graded algebras, and explain their relationship with forms on path spaces.