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Natural operators lifting vector fields to bundles of Weil contact elements

Miroslav Kureš, Włodzimierz M. Mikulski (2004)

Czechoslovak Mathematical Journal

Let A be a Weil algebra. The bijection between all natural operators lifting vector fields from m -manifolds to the bundle functor K A of Weil contact elements and the subalgebra of fixed elements S A of the Weil algebra A is determined and the bijection between all natural affinors on K A and S A is deduced. Furthermore, the rigidity of the functor K A is proved. Requisite results about the structure of S A are obtained by a purely algebraic approach, namely the existence of nontrivial S A is discussed.

Newforms, inner twists, and the inverse Galois problem for projective linear groups

Luis V. Dieulefait (2001)

Journal de théorie des nombres de Bordeaux

We reformulate more explicitly the results of Momose, Ribet and Papier concerning the images of the Galois representations attached to newforms without complex multiplication, restricted to the case of weight 2 and trivial nebentypus. We compute two examples of these newforms, with a single inner twist, and we prove that for every inert prime greater than 3 the image is as large as possible. As a consequence, we prove that the groups PGL ( 2 , 𝔽 ℓ 2 ) for every prime ( ℓ ≡ 3 , 5 ( mod 8 ) , ℓ > 3 ) , and PGL ( 2 , 𝔽 ℓ 5 ) for every prime ℓ ¬ ≡ 0 ± 1 ( mod 11 ) ; ℓ > 3 ) , are Galois groups...

Niven’s Theorem

Artur Korniłowicz, Adam Naumowicz (2016)

Formalized Mathematics

This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

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