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Question:

are there any "standard" ways of indicating the meaning of numerical values in formulas, resp. general mathematical texts (theorems, proofs, etc.)?

I am especially looking for suggestions to tell apart the weight of a graph's edge from a multiplicity of that weight.
Background is that I want to demonstrate, how to calculate a sufficiently high constant that, when added or assigned to certain edge's weights, does some "tricks" e.g., ensuring that some specific edges are included in the output of a tsp solver.

As an example, consider the case where I have e.g. 5 edges, each of weight $1$ and I want to demonstrate via some calculations how I reach a result, but want to allow for a clear distinction between the cardinal 5 and the scalar weight $1$. Is it in that case customary to use different fonts like $5*\mathbf{1}$ or rather to put the cardinals to one side of the multiplication sign and the weight on the other?

I am aware that what is considered "good" or "best" is largely in the eye of the beholder, so this question is aimed at an overview of notations.
Some background on the origin and/or articles in which a specific notation is used would be great.

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    $\begingroup$ Maybe you can give an example of what you intend. For starters, there's always good old \underbrace... $\endgroup$ Commented Apr 18, 2017 at 5:47
  • $\begingroup$ @NateEldredge example added; I hope it suffices to clarify, what I have in mind. $\endgroup$ Commented Apr 18, 2017 at 8:33
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    $\begingroup$ I'd write something like $w_1n_1 + w_2n_2 = 1 \cdot 5 + 2\cdot 3 = 11$. It's now clear that $w_i$ are the weights, $n_i$ are the counts. Once you have the symbolic formula written down, substituting numbers in it doesn't matter much to a mathematician, usually. $\endgroup$ Commented Apr 18, 2017 at 11:39

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Two ways to indicate a rescaling of the weight of the edges of a graph, by an additive constant or by an arbitrary multiple.

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  • $\begingroup$ Actually, I have no illustration and the case of counting the multiplicity of edges with the same weight doesn't occur in the illustrations (which are nonetheless related to my question and useful). $\endgroup$ Commented Apr 18, 2017 at 8:36
  • $\begingroup$ added an example of the use of weight rescaling factor $\alpha$ in a formula (from the same source); in response to your example, I would just write $5\alpha$ for the general case and specify $\alpha=1$ for the special case of unit rescaling factor. $\endgroup$ Commented Apr 18, 2017 at 9:37

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