RECOMMENDATIONS TO APPLIED REAL ANALYSIS

 Newest

BASS,, R.; Real Analysis for Graduate Students, Secod Edition; CreateSpace Independent Publishing Platform, 418p. 2013

                        Classics

APOSTOL, T.; Mathematical Analysis, Pearson, 492p., 1974

BARTLE, R.; Sherbert, D.; Introduction to Real Analysis; Wiley, 416p., 2011

GARDING, L.; Encounter with mathematics, Springer-Verlag, 270p., 1977

KOLMOGOROV, A.; FOMIN, S.; SILVERMAN, R.; Introdutory Real Analysis, Dover Publications, 416p., 1975

RUDIN, W.; Principles of Mathematical Analysis; McGraw-Hill, 325p., 1976

WITTGENSTEIN, L.; Tractatus Logico-Philosophicus, CreateSpace Independent Publishing Plataform, 96p., 2011

For Brazilians: challenging, intensive and provocative

LIMA, E.; Análise Real – volume 1: Funções de uma Variável, 11ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Matemática Universitária, 198p., 2012

LIMA, E.; Análise Real – volume 2: Funções n Variáveis, 5ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Matemática Universitária, 209p., 2010

LIMA, E.; Curso de Análise vol. 1, 14ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Projeto Euclides, 431p., 2013

LIMA, E.; Curso de Análise vol. 2, 11ª edição, Rio de Janeiro: IMPA - Instituto de Matemática Pura e Aplicada, Coleção Projeto Euclides, 547p., 2012

                       Applied Real Analysis

OK, E.; Real Analysis with Economic Applications; Princeton University Press, 664p. 2007

(We intend to use this book on 2015 Summer Course)

Questions:

The aim of this first chapter is to provide the necessary elements for understanding the structure of theoretical models in economics. In a point of view, study the models and their detailed assumptions. So, based in a framework we could test, corroborate and infer the theoretical structure thereof.  

Questions:

What’s the differences amongst theorem, lemma, corollary, and axiom?

Why there exists?

Why we need to prove them or take some of them as truth?

Theorem: a theorem in a theoretical model is a statement that has been proven on the basis of theoretical foundations and previous concepts, such as other theorems, corollaries, axioms, and lemmas. For instance, the Nash Theorem.

Lemma: is a “helping theorem”, a proposition with little applicability except to be a part of the proof of a larger theorem. In some cases, as in Finance  we could cite the Ito’s Lemma.

Corollary: is a proposition that follows with little or no proof from one other theorem or definition. Example: Walras’ Corollary in General Equilibrium.

Axiom: considerate a postulate is a statement that is accepted without proof and regarded as fundamental to a subject. The main example is the Probability Axioms.

Note: one of the most interesting articles that shows the importance of ability in mathematics was written by Franklin Joel, entitled “Mathematical Method of Economics”, published by The American Mathematical Monthly in 1983. In that paper, the author emphasizes the role of mathematics in academic work of prized economists.

            It’s also interesting to stress that, in the nineteenth century, a scholar or noble person was recognized for his skills in chess and mathematics. Seeing mathematics as a major obstacle to Lord Keynes, in the twentieth century, he wrote to another economist complaining about studies in math.