Liouville numbers, Liouville sets and Liouville fields
نویسندگان
چکیده
منابع مشابه
Liouville numbers
In this work, we define the concept of Liouville numbers as well as the standard construction to obtain Liouville numbers and we prove their most important properties: irrationality and transcendence. This is historically interesting since Liouville numbers constructed in the standard way where the first numbers that were proven to be transcendental. The proof is very elementary and requires on...
متن کاملLiouville Closed H-fields
H-fields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending R and fields of transseries over R are H-fields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed H-fields, and study various constructions in the category of H-fields: closure under powers, constant field extension, completion, and buil...
متن کاملAll Liouville Numbers are Transcendental
In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http: //www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an ...
متن کاملH-fields and Their Liouville Extensions
We introduce H-fields as ordered differential fields of a certain kind. Hardy fields extending R, as well as the field of logarithmicexponential series over R are H-fields. We study Liouville extensions in the category ofH-fields, as a step towards a model theory ofH-fields. The main result is that an H-field has at most two Liouville closures.
متن کاملThe Kolmogorov Complexity of Liouville Numbers
We consider for a real number α the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and self-delimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative information of one base with respect to the other. More...
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ژورنال
عنوان ژورنال: Proceedings of the American Mathematical Society
سال: 2015
ISSN: 0002-9939,1088-6826
DOI: 10.1090/proc/12408