Global Dimensional Mathematics

Here we build the fundamentals of global dimensional mathematics in order to build the new basis of the new theoretical scientific paradigm. Research on the foundations of mathematics covering the definition of fundamental mathematical concepts such as point, line, direction, dimension, addition, multiplication, division, zeros, infinities, limits, factorization, integers and prime numbers, were carried out and further still the resolution of the Goldbach conjecture is now effective. New original fundamental mathematical notions are established building the core of global dimensional mathematics based on the decomposition of integers into an addition of prime numbers terms and on fundamental geometric concepts such as the concept dimension based on the notion of direction and point, as well as research on set theory and work on the notion of limits and infinity. The termization as a decomposition of integers into an addition of prime number terms by a python program, breaks the unique factorization theorem, the fundamental theorem of arithmetic while the geometric notions developed break Euclidean geometry leading to a new mathematical framework, geometry, topology and metrics leading to a total change of theoretical scientific paradigm. Global dimensional mathematics forms the basis for the construction of the new scientific paradigm of the 21st century and beyond, opening up a still unknown perspective on the world of science in general.

A construção do conhecimento matemático na teoria APOS

ResumoEste artigo é um recorte de uma tese de doutorado em andamento, desenvolvida pela primeira autora e orientada pela segunda, cujo objetivo principal é investigar a concepção, na acepção de Dubinsky, de professores do ensino público paulista sobre o Teorema Fundamental da Aritmética. O objetivo deste artigo é trazer a ideia de como ocorre a construção do conhecimento matemático segundo a teoria APOS de Ed Dubinsky, apoiado nas ideias de Jean Piaget com a Epistemologia Genética. São apresentados o conceito de abstração reflexiva, considerado o mecanismo mental mais importante no desenvolvimento do pensamento, as definições dos elementos mentais Ação, Processo e Objeto e como interagem entre si para a construção do conhecimento matemático. Palavras-chave: Construção do conhecimento matemático; Teoria APOS; abstração reflexiva; ciclo ACE.AbstractThis article is an excerpt from an ongoing doctoral thesis, developed by the first author and guided by the second, whose main objective is to investigate the conception, in Dubinsky's sense, of São Paulo public school teachers about the Fundamental Theorem of Arithmetic. The purpose of this article is to bring the idea as is the construction of mathematical knowledge according to APOS theory of Ed Dubinsky, supported by the Jean Piaget's ideas with the Genetic Epistemology. The concept of reflective abstraction is presented, considered the most important mental mechanism in the development of thought, the definitions of the mental elements Action, Process and Object and how they interact with each other for the construction of mathematical knowledge.Keywords: Construction of mathematical knowledge; APOS theory; reflective abstraction; ACE cycle.

Introducing the fundamental theorem of arithmetic through a mobile game

There is an increasing trend towards the use of mobile games in education. Presenting knowledge with mobile games requires many variables to be employed. These processes should be made more rigorous in domains such as mathematics where knowledge is abstract. The aim of this study is to develop an application to introduce the Fundamental Theorem of Arithmetic through a mobile game. The findings of the research show that the Fundamental Theorem of Arithmetic can be introduced with the developed mobile game. When the mobile game is evaluated in terms of mathematical knowledge, it is determined that while the constraints and conditions determined in the game hide the mathematical knowledge. In this sense, some game examples are given in this study and some models related to feedbacks that students can take in these games are presented. Keywords: a-didactical situations, mobile games, mathematics teaching, mathematical concepts;

The algorithm

Mathematics has formed a partnership with computer science, since at least the 1940s. Devising algorithms and turning them into computer programs is akin to deconstructing a problem, a proof, or a theory into its key components. In other words, a computer can be used to attack the problem of decidability, discussed in the previous chapter, on the basis of the validity (or not) of some algorithm. This chapter deals with this key notion, completing the chain of ideas described in this book that started with Pythagoras. Although not named in this way, the concept of algorithms goes back to Euclid and his Fundamental Theorem of Arithmetic. Computers are adept at solving all kinds of problems that involve pattern and structure. They do so by computing all possibilities for a problem, rather than providing a theoretical explanation for the solution. The latter is left up to the programmer—in this case, the mathematician.

The Fundamental Theorems of Hyper-Operations

We examine the extensions of the basic arithmetical operations of addition and multiplication on the natural numbers into higher-rank hyper-operations also on the natural numbers. We go on to define the concepts of prime and composite numbers under these hyper-operations and derive some results about factorisation, resulting in fundamental theorems analogous to the Fundamental Theorem of Arithmetic.

Construction of Quasi-Cyclic LDPC Codes Based on Fundamental Theorem of Arithmetic

Quasi-cyclic (QC) LDPC codes play an important role in 5G communications and have been chosen as the standard codes for 5G enhanced mobile broadband (eMBB) data channel. In this paper, we study the construction of QC LDPC codes based on an arbitrary given expansion factor (or lifting degree). First, we analyze the cycle structure of QC LDPC codes and give the necessary and sufficient condition for the existence of short cycles. Based on the fundamental theorem of arithmetic in number theory, we divide the integer factorization into three cases and present three classes of QC LDPC codes accordingly. Furthermore, a general construction method of QC LDPC codes with girth of at least 6 is proposed. Numerical results show that the constructed QC LDPC codes perform well over the AWGN channel when decoded with the iterative algorithms.