Um Irracional: O Número de Euler

http://dx.doi.org/10.5902/2179460X14244The set of real numbers, R, is a delicate issue to be addressed in the classroom. In particular, the study of the set of irrational numbers is hampered by lack of examples found in many textbooks. The main goal of this paper is presenting of clear manner an important example of irrational number: Euler’s number. Here we present the motivation for the Euler’s number as well, the proof of their irrationality.O conjunto dos números reais, R, é um tema delicado de ser tratado na sala de aula. Em particular, o estudo do conjunto dos números irracionais fica prejudicado pela falta de exemplos encontrada em muitos livros didáticos. O principal objetivo deste trabalho é apresentar de maneira clara um importante exemplo de número irracional: o número de Euler. Aqui apresentaremos a motivação para o surgimento do número de Euler, bem como, a demonstração de sua irracionalidade

Book of Abstracts

USPCAPESFAPESPCNPqINCTMatICMC Summer Meeting on Differentail Equations.\ud São Carlos, Brasil. 3-7 february 2014

C(k)-solvability near the characteristic set for a class of planar complex vector fields of infinite type

This paper deals with semi-global C(k)-solvability of complex vector fields of the form L = partial derivative/partial derivative t + x(r) (a(x) + ib(x))partial derivative/partial derivative x, r >= 1, defined on Omega(epsilon) = (-epsilon, epsilon) x S(1), epsilon > 0, where a and b are C(infinity) real-valued functions in (-epsilon, epsilon). It is shown that the interplay between the order of vanishing of the functions a and b at x = 0 influences the C(k)-solvability at Sigma = {0} x S(1). When r = 1, it is permitted that the functions a and b of L depend on the x and t variables, that is, L = partial derivative/partial derivative t + x(a(x, t) + ib(x, t))partial derivative/partial derivative x, where (x, t) is an element of Omega(epsilon).FAPESPFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP

Resolubilidade global para uma classe de campos vetoriais no Toro.

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Solvability near the characteristic set for a special class of complex vector fields

This work deals with the solvability near the characteristic set Sigma = {0} x S-1 of operators of the form L = partial derivative/partial derivative t+(x(n) a(x)+ ix(m) b(x))partial derivative/partial derivative x, b not equivalent to 0 and a(0) not equal 0, defined on Omega(epsilon) = (-epsilon, epsilon) x S-1, epsilon > 0, where a and b are real-valued smooth functions in (-epsilon, epsilon) and m >= 2n. It is shown that given f belonging to a subspace of finite codimension of C-infinity (Omega(epsilon)) there is a solution u is an element of L-infinity of the equation Lu = f in a neighborhood of Sigma; moreover, the L-infinity regularity is sharp.CNPqFAPES

Solvability of a first order differential operator on the two-torus

Global solvability on the two-torus of a first order differential operator with complex coefficients is investigated. Diophantine properties of the coefficients are linked to the solvability.CNPqFAPES