Souvenirs from Bandung part 1 (MAC)

 

whiteboard at MAC

 

MAC, stands for Mathematics Aid Center, is a consultation room in Mathematics Study Program of ITB. If you’re an ITB student and have problems regarding Mathematics, especially problems from courses, you are welcomed to come and ask for assistance. I used to manage this room for a long time. Too long perhaps, for my name is clearly associated with MAC, as well as MAC’s name is associated to me 😆 Anyway, I had an opportunity to visit this nostalgic room yesterday, met the new coordinator, as well as some of the assistants.

The moment I stepped in, Akbar, an assistant in MAC and a good friend welcomed me by presenting his Master thesis problem right away. He clearly didn’t miss me that much, did he? 😀 Before I start with his problem, let me give you a short escort to it. If we talk about metric spaces, and if we unfortunately luckily have two metrics on it, we would like to discuss the relationship between them. For instance, if f is a continuous function with respect to the first metric d_1, is it also continuous with respect to the second metric d_2? One pretty ideal relationship between two metrics are equivalence. Metric d_1 is equivalent with metric d_2 if there’s constants m, M such that md_1 < d_2 < Md_1.

If a sequence converges on d_2, then by the first inequality, it also converges on d_1. The other way is also true by the second inequality. Since theory of converging sequences is one of the main building blocks of mathematical analysis, we can say so many things just by having this equivalence. If you know how continuous functions can be defined by the use of converging sequences, then given two equivalent metrics, you can answer the question above with “yes”. Anyway, I try my best not to go too much into details and definition here, so if you wanna know more about the term equivalence of metrics please resort to google.

Akbar’s thesis deal with the metric defined by the p-norm:

\|\mathbf{x}\|_p := \bigg( \sum_{i=1}^{\infty} |x_i|^p \bigg)^{1/p}.

And he wants to shows the equivalence between that metric with the following:

\|\mathbf{x},\mathbf{y}\| := \sup_{\|\mathbf{w}\|_p, \|\mathbf{z}\|_p \leq 1} \sum_{i=1}^{\infty} \sum_{j=1}^{\infty} \begin{pmatrix} x_i & x_j\\y_i & y_j \end{pmatrix} \begin{pmatrix} w_i & w_j\\z_i & z_j \end{pmatrix}.

Taking the first metric (from the p-norm) as d_2 and the the the second one as d_1 and then look back at our lousy definition of equivalence, Akbar still have to prove the second inequality. I heard the first inequality has already been proven by his supervisor and became a paper by itself. CMIIW. So, we can approximate the difficulty to establish that inequality huh. What a warm welcome you gave me bar 😛 Anyone want to ingenuously help him with this? 😀

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