This is a question about math notation. The context is, I'm trying to correctly define a problem of scheduling the flow through a set of devices, $D = \{d_1, d_2, ... d_n\}$, over a finite discrete future planning horizon $T$ = $(1,2,3\dots,60)$. The schedule of all devices over the planning horizon can be described by a real valued matrix, $Q$, with $|D|$ rows and $|T|$ columns. To define $Q$ I could state just that. But given a definition of $D$ and $T$ which form the "index sets", or just "indices" of matrix, to define $Q$ could/should I just say "Let $Q = D \times T \rightarrow \mathbb{R}$", or something else that is more succinct and direct than "Let $Q$ be a matrix with $|D|$ rows and $|T|$ columns"?
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$\begingroup$ This depends a bit on the intended audience. Sometimes having a super technical definition isn't as useful as a more clear but less precise one. Perhaps the shortest commonly used notation would be $Q\in\mathbb{R}^{|D|\times|T|}$ (or $Q\in\mathbb{R}^{m\times n}$ if you defined $m$ to be the size of $T$). $\endgroup$– overfull hboxJan 16, 2019 at 4:06
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Is there any advantage to representing your flows as a matrix ? Will you be adding sets of flow values from different scenarios, for example ?
If not, you might as well just define a function $f:D \times T \rightarrow \mathbb{R}$ where $f(d_i, t)$ is the flow through device $d_i$ at time $t$. And you might want to think about whether your domain is really the whole of $\mathbb{R}$. For example, can flow values be negative ?
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$\begingroup$ Hmm interesting idea. I do need to define the matrix though as I need to define constraints on the joint flows. Adding is another potential use case. Yes the flows can be negative some flows in are related to flows out. I now know the class of problem is usually called a "network flow problem" popular in traffic routing for instance, except that it is less common to include the time domain, as above. $\endgroup$– spinkusJan 16, 2019 at 21:46