What are possible ways to say "represented by" or "characterised by" in Mathematics? e.g. When a Mathematical object is used to represent or characterise a physical quantity or some mathematical object from a different field:
- In Quantum Mechanics: $\doteq$ or $ \equiv$
- In representation theory (matrices representing other mathematical objects).
Is there a universal notation (or notations) proposed which can be used across various fields?
It may be used to relate objects from a different field (physics, etc) to Mathematics, or algebraic notation, or simply a symbolic "notation".
The best I could find so far is $\doteq$ (See below; Sakurai & Napolitano 2nd Ed, page 23, also see ). Others authors have used notations for this purpose for similar reasons.
$$\mid +\rangle\doteq\begin{pmatrix}0\\1\end{pmatrix}$$
I thought maybe there are notations for "represented by" in following areas that can be used outside their context. I think it is useful to know if there are other instances of such usage in following fields:
- Isomorphism may imply representations (in representation theorems). In general (e.g for reducible representations) a "represented by" relation may be a homeomorphism or even no morphism (numerical approximation).
- Certain arrow symbols in category theory
- In probability theory, when a random variable is linked to a distribution: $\thicksim$: e.g. $X_i\thicksim N(\mu_i,\sigma)$
- Approximations in numerical analysis: $\tilde{x}\approx x$, or $\tilde{x_1}=x_1\pm\varepsilon_1$.
- In general, a definition or equality can be seen as a "representation" relationship:
$:=$ or $\overset{\mathrm{def}}{==}$ or $=$.
But above usages do not imply the "represented by" or "characterised by" explicitly. They do so only implicitly, or from the context or the text.
Best Answer
I think that there is a reason that no one has jumped in to answer your question in the way you were perhaps hoping for.
As your examples suggest, "represented" can mean quite different things in different contexts.
Going along with your examples for the sake of argument, we can all surely agree that
are very different enterprises, and only confusion could possibly arise from assimilating them too closely. So why should we even want a common notation? Isn't part of the point of a good notational system to mark important commonalities and important differences.
The very fact that you struggle to give a generic definition of "representation" which smoothly covers all your cases suggests that, no, we don't have a single notion at work in all these cases. So there's no reason to want a single notation.