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A note on characterizations of rings of constants with respect to derivations

Piotr Jędrzejewicz (2004)

Colloquium Mathematicae

Let A be a commutative algebra without zero divisors over a field k. If A is finitely generated over k, then there exist well known characterizations of all k-subalgebras of A which are rings of constants with respect to k-derivations of A. We show that these characterizations are not valid in the case when the algebra A is not finitely generated over k.

A note on finitely generated ideal-simple commutative semirings

Vítězslav Kala, Tomáš Kepka (2008)

Commentationes Mathematicae Universitatis Carolinae

Many infinite finitely generated ideal-simple commutative semirings are additively idempotent. It is not clear whether this is true in general. However, to solve the problem, one can restrict oneself only to parasemifields.

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

If X is a smooth scheme over a perfect field of characteristic p , and if 𝒟 X ( ∞ ) is the sheaf of differential operators on X [7], it is well known that giving an action of 𝒟 X ( ∞ ) on an 𝒪 X -module ℰ is equivalent to giving an infinite sequence of 𝒪 X -modules descending ℰ via the iterates of the Frobenius endomorphism of X [5]. We show that this result can be generalized to any infinitesimal deformation f : X → S of a smooth morphism in characteristic p , endowed with Frobenius liftings. We also show that it extends to adic...

A note on the pp conjecture for sheaves of spaces of orderings

Paweł Gładki (2016)

Communications in Mathematics

In this note we provide a direct and simple proof of a result previously obtained by Astier stating that the class of spaces of orderings for which the pp conjecture holds true is closed under sheaves over Boolean spaces.

A p-adic behaviour of dynamical systems.

Stany De Smedt, Andrew Khrennikov (1999)

Revista Matemática Complutense

We study dynamical systems in the non-Archimedean number fields (i.e. fields with non-Archimedean valuation). The main results are obtained for the fields of p-adic numbers and complex p-adic numbers. Already the simplest p-adic dynamical systems have a very rich structure. There exist attractors, Siegel disks and cycles. There also appear new structures such as fuzzy cycles. A prime number p plays the role of parameter of a dynamical system. The behavior of the iterations depends on this parameter...

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