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If I come across something cool, I'll try to write about it over here. Each post is sorted into categories (check the about page).

Warning

For STEM stuff especially, I try my best with research, but I am no professional so there may be some inaccuracies here and there with what I say.

Recent Posts :

  • in Maths
  • 11 min read

An "Almost Always" Approach to Sequences

Introduction

A sequence in some space \(X\) is usually represented by some function \(a:\mathbb{N}\to X\) so that the first term of the sequence is \(a(1)\), the second is \(a(2)\) and so on. However, indices are often used to represent these terms, i.e. we usually write \(a_n\) in place of \(a(n)\). Keep in mind that we shall switch between these two conventions where appropriate.

  • in Maths
  • 3 min read

A technique for constructing subsequences

Say we are given a sequence \((a_n)\) from which we would like to construct a subsequence \((a_{n_k})\). We can think of each \(a_n\) as a fish flowing down a river, and they do so in sequence; first comes the fish \(a_1\), then \(a_2\), and so on. The idea here is to set up a "fishing net". Depending on the net, some fish can easily swim through while others may get caught. As the sequence of fish \(a_1, a_2, \ldots\) attempt to swim through the net, we note down each time a fish gets caught and that is how we construct our subsequence. Of course, in order for all of this to work we have to make sure that our "net" does not eventually run out of fish to catch.

  • in Maths
  • 8 min read

Sums over Countable Spaces

The Finite Case

Suppose I have a space \(X\) with three elements, we can represent this space with a set \(X:=\{\alpha, \beta, \gamma\}\). I want to be able to talk about sums over this space. Of course, \(\alpha + \beta + \gamma\) doesn't make any sense. The elements of \(X\) simply serve as placeholders for the values that we want to add, in other words, we are using the "three-ness" of the space \(X\) to define a sum of three values. We can do this by assigning to each element \(x\in X\) a real number \(r\in \mathbb{R}\), i.e. by defining a weight function \(f:X\to \mathbb{R}\). For example, if we define \(f\) such that \(f(\alpha) = f(\beta) = 3\) and \(f(\gamma)=10\), then it makes sense to proceed as follows :

\[ \begin{align*} \sum_{x\in X}f(x) &= f(\alpha) + f(\beta) + f(\gamma) \\ &= 3 + 3 + 10 \\ &= 16 \end{align*} \]