One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

Submitted by Tassia Roberta Zangiacomo null (tassia_zangiacomo@hotmail.com) on 2017-03-23T22:04:31Z No. of bitstreams: 1 TASSIA ROBERTA ZANGIACOMO - MESTRADO.pdf: 1004175 bytes, checksum: 12925ba240f8d9a89e295b32b2efb13e (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-24T17:23:14Z (GMT) No. of bitstreams: 1 zangiacomo_tr_me_rcla.pdf: 1004175 bytes, checksum: 12925ba240f8d9a89e295b32b2efb13e (MD5) / Made available in DSpace on 2017-03-24T17:23:15Z (GMT). No. of bitstreams: 1 zangiacomo_tr_me_rcla.pdf: 1004175 bytes, checksum: 12925ba240f8d9a89e295b32b2efb13e (MD5) Previous issue date: 2017-02-20 / Este trabalho tem como objetivo construir os sistemas numéricos usuais, a saber, o conjunto dos números naturais N, o conjunto dos números inteiros Z, o conjunto dos números racionais Q e o conjunto dos números reais R. Iniciamos o trabalho tratando de noções sobre conjuntos e relações binárias. Em seguida, apresentamos o conjunto dos números naturais, definido através dos axiomas de Peano; o conjunto dos números inteiros via uma relação de equivalência com o conjunto dos números naturais; o conjunto dos números racionais, que são obtidos também via relação de equivalência, mas dessa vez com o conjunto dos números inteiros; a construção do conjunto dos números reais, feita via cortes no conjunto dos números racionais; e, para todos esses casos, mostramos a imersão do conjunto anterior no conjunto que surge na sequência. Por fim, observamos alguns materiais do ensino fundamental e médio com o intuito de investigar de que forma esses temas estão sendo apresentados para os alunos. / This work aims to construct the usual numerical systems, namely the set of natural numbers N, the set of integers Z, the set of rational numbers Q and the set of real numbers R. We begin the work dealing with notions about sets and binary relations. Next, we present the set of natural numbers, defined by Peano's axioms; the set of integers via an equivalence relation with the set of natural numbers; the set of rational numbers, which are also obtained via equivalence relation, but this time with the set of integers; the construction of the set of real numbers, made through cuts in the set of rational numbers; end for all these cases we show the immersion of the previous set in the ensemble that appears in the sequence. Finally, we observed some materials in elementary school and high school in order to investigate how these themes are being presented to the students.

We seek to discover combinatorial explanations of known identities involving harmonic numbers. Harmonic numbers do not readily lend themselves to combinatorial interpretation, since they are sums of fractions, and combinatorial arguments involve counting whole objects. It turns out that we can transform these harmonic identities into new identities involving Stirling numbers, which are much more apt to combinatorial interpretation. We have proved four of these identities, the first two being special cases of the third.

No description available.

A long standing unanswered question in number theory concerns the existence (or not) of odd perfect numbers. Over time many properties of an odd perfect number have been established and refined. The initial goal of this research was to improve the lower bound on the number of distinct prime factors of an odd perfect number, if one exists, to at least 9. Previous approaches to this problem involved the analysis of a carefully chosen set of special cases with each then being eliminated by a contradiction. This thesis applies an algorithmic, factor chain approach to the problem. The implementation of such an approach as a computer program allows the speed, accuracy and flexibility of modern computer technology to not only assist but even direct the discovery process. In addition to considering odd perfect numbers, several related problems were investigated, concerned with (i) harmonic, (ii) even multiperfect and (iii) odd triperfect numbers. The aim in these cases was to demonstrate the correctness and versatility of the computer code and to fine tune its efficiency while seeking improved properties of these types of numbers. As a result of this work, significant improvements have been made to the understanding of harmonic numbers. The introduction of harmonic seeds, coupled with a straightforward procedure for generating most harmonic numbers below a chosen bound, expands the opportunities for further investigations of harmonic numbers and in particular allowed the determination of all harmonic numbers below 10 to the power 12 and a proof that there are no odd harmonic numbers below 10 to the power 15. When considering even multiperfect numbers, a search procedure was implemented to find the first 10-perfect number as well as several other new ones. As a fresh alternative to the factor chain search, a 0-1 linear programming model was constructed and used to show that all multiperfect numbers divisible by 2 to the power of a, for a being less than or equal to 65, subject to a modest constraint, are known in the literature. Odd triperfect numbers (if they exist) have properties which are similar to, but simpler than, those for odd perfect numbers. An extended test on the possible prime factors of such a number was developed that, with minor differences, applies to both odd triperfect and odd perfect numbers. When applicable, this test allows an earlier determination of a contradiction within a factor chain and so reduces the effort required. It was also shown that an odd triperfect number must be greater than 10 to the power 128. While the goal of proving that an odd perfect number must have at least 9 distinct prime factors was not achieved, due to mainly practical limitations, the algorithmic approach was able to show that for an odd perfect number with 8 distinct prime factors, (i) if it is exactly divisible by 3 to the power of 2a then a = 1, 2, 3, 5, 6 or a is greater than or equal to 31 (ii) if the special component is pi to the power of alpha, pi less than 10 to the 6 and pi to the (alpha+1) less than 10 to the 40, then alpha = 1.

Thesis (doctoral)--Universität Marburg, 1911. / Vita. Includes bibliographical references.

Thesis--Copenhagen. / "Fortegnelse over større værker, som gentagne gange citeres": p. [x].

published_or_final_version / Mathematics / Master / Master of Philosophy

The purpose of this paper is to review and summarize the topics involved in the study of real analysis. Real analysis is a branch of mathematics that studies the field of real numbers including the calculus of real numbers, analytical properties of real functions and sequences. This includes limits of sequences of real numbers, continuity, completeness, and related properties of real functions. While all topics in the course were important and vital to understanding analysis, the goal of this paper is to review, research, and report on a few of the more interesting topics covered in the class. / text

published_or_final_version / Mathematics / Master / Master of Philosophy