What does "$\sim$" mean here? $F(w_i-w_j,\tilde w_k)=\frac{P_{ik}}{P_{jk}}$

Question

What does this "$\sim$" symbol mean? For example, $$F(w_i-w_j,\tilde w_k)=\frac{P_{ik}}{P_{jk}}$$

Answer 1

It varies according to context.

Sometimes it's used to denote another, related concept, variable, etc., to the one without the tilde; for instance, it might help readers to note that $w$ and $x$ are similar in structure, so $x$ would be written as $\tilde w$.

Sometimes it is used as a binary operation; for example, $a\sim b$ if and only if $a$ and $b$ are in the same coset of a quotient group of a group.

Sometimes it's used above equality signs to denote some weaker sense of equality, like an isomorphism; for example, $A\cong B$ for groups $A, B$ means "$A$ is isomorphic to $B$ as a group."

Doubtless, there are countless many other uses too.