Year: 2026
Type: article
Abstract: Given 𝑏0 ∈ R and 𝑐0 ∈ R, let the families of functions 𝐹a : R → R be given by 𝐹a (𝑥) = 𝑎𝑥2 + 𝑏0x+ c0 , 𝑎 ∈ R*, and𝑓a : C → C be given by 𝑓a (𝑧) = 𝑎𝑧2 + 𝑏0z+ c0 , 𝑎 ∈ R*. When we vary 𝑎, we obtain the FaM-representation, which consists of the Cartesian plane in which the graph of 𝐹a evolves and, right below, of the complex plane in which the paths of the 2 roots of 𝑓a are traced. The importance of the FaM-representation lies in the simultaneous analysis of what happens to 𝐹a with what happens to the 2 roots of 𝑓a. We show that the sequences of functions 𝐹n + -1 : [0,+∞[→ [0,+∞[ given by 𝐹n + -1(𝑊) = (𝑊/𝑛)1/2 and 𝐹n - -1 : [0,+∞[→] − ∞, 0] given by 𝐹n - -1(𝑊) = - (𝑊/𝑛)1/2, 𝑛 ∈ N = {1, 2, 3, ...}, converge pointwise, but not uniformly, to 0; while 𝐹n + -1|[0,M] and 𝐹n - -1|[0,M] converge uniformly to 0, for fixed 𝑀 > 0. For each 𝑎 ∈ R*, we determine the curvature of 𝜆a : R → R2 given by 𝜆a(𝑥) = (𝑥, a𝑥2) at a fixed 𝑥0, and give examples.
Author: Luís Cláudio Yamaoka
Institution: Universidade Federal de São Paulo
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W7222932915