Version 1
: Received: 19 July 2020 / Approved: 19 July 2020 / Online: 19 July 2020 (15:28:45 CEST)
Version 2
: Received: 4 August 2020 / Approved: 5 August 2020 / Online: 5 August 2020 (05:43:50 CEST)
Version 3
: Received: 18 September 2021 / Approved: 20 September 2021 / Online: 20 September 2021 (15:11:14 CEST)
Version 4
: Received: 13 August 2022 / Approved: 15 August 2022 / Online: 15 August 2022 (04:55:03 CEST)
How to cite:
Ramírez, J. Logarithmic-Time Addition for BIT-Predicate With Applications for a Simple and Linear Fast Adder And Data Structures. Preprints2020, 2020070415. https://doi.org/10.20944/preprints202007.0415.v4
Ramírez, J. Logarithmic-Time Addition for BIT-Predicate With Applications for a Simple and Linear Fast Adder And Data Structures. Preprints 2020, 2020070415. https://doi.org/10.20944/preprints202007.0415.v4
Ramírez, J. Logarithmic-Time Addition for BIT-Predicate With Applications for a Simple and Linear Fast Adder And Data Structures. Preprints2020, 2020070415. https://doi.org/10.20944/preprints202007.0415.v4
APA Style
Ramírez, J. (2022). Logarithmic-Time Addition for BIT-Predicate With Applications for a Simple and Linear Fast Adder And Data Structures. Preprints. https://doi.org/10.20944/preprints202007.0415.v4
Chicago/Turabian Style
Ramírez, J. 2022 "Logarithmic-Time Addition for BIT-Predicate With Applications for a Simple and Linear Fast Adder And Data Structures" Preprints. https://doi.org/10.20944/preprints202007.0415.v4
Abstract
A construction for the systems of natural and real numbers is presented in Zermelo-Fraenkel Set Thoery, that allows for simple proofs of the properties of these systems, and practical and mathematical applications. A practical application is discussed, in the form of a Simple and Linear Fast Adder (Patent Pending). Applications to finite group theory and analysis are also presented. A method is illustrated for finding the automorphisms of any finite group $G$, which consists of defining a canonical block form for finite groups. Examples are given, to illustrate the procedure for finding all groups of $n$ elements along with their automorphisms. The canonical block form of the symmetry group $\Delta_4$ is provided along with its automorphisms. The construction of natural numbers is naturally generalized to provide a simple and sound construction of the continuum with order and addition properties, and where a real number is an infinite set of natural numbers. A basic outline of analysis is proposed with a fast derivative algorithm. Under this representation, a countable sequence of real numbers is represented by a single real number. Furthermore, an infinite $\infty\times\infty$ real-valued matrix is represented with a single real number. A real function is represented by a set of real numbers, and a countable sequence of real functions is also represented by a set of real numbers. In general, mathematical objects can be represented using the smallest possible data type and these representations are calculable. In the last section, mathematical objects of all types are well assigned to tree structures in a proposed type hierarchy.
Structuralism; Set Theory; Type Theory; Arithmetic Model; Data Type; Tree, Group
Subject
Computer Science and Mathematics, Logic
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received:
15 August 2022
Commenter:
Juan Ramirez
Commenter's Conflict of Interests:
Author
Comment:
The sections on Set Theory have been updated. This includes the subsections Motivation and Formalization. The arithmetic model proposed is used as basis for a SImple and Linear Fast Adder (Patent Pending) which was not included in previous versions. The section on finite groups which describes a canonical block form has been made shorter and an appendix with examples is included.
Commenter: Juan Ramirez
Commenter's Conflict of Interests: Author