1

I'm writing research that involves explaining objects which are fairly complicated and very specific to the research in question (e.g. a new type of mathematical model of something). The objects in question may have complicated constraints or methods of construction that need explaining. Such explanations may require multiple paragraphs/sentences with a lot of notation. I want to be clear that these explanations apply to the object in a general sense (applies to any object of this type) rather than a specific instance of this object. But achieving this becomes difficult because the writing starts to become ambiguous or awkward as a result of the object's complicated nature.

Here is a simple example which I will make up for the purposes of this question:

Here, an index matrix is a matrix of positive integers that is associated with a $N$ value and a $K$ value. Each row of an index matrix $X$ is a permutation of the smallest $K(X)$ positive integers, where $K(X)$ is $X$'s $K$ value. There cannot exist a positive integer which is equal to 2 or more elements of a row in an index matrix. Hence, the number of columns in an index matrix $X$ must be less than or equal to $K(X)$. The number of rows in an index matrix $X$ is $2^{N(X)}$ where $N(X)$ is $X$'s $N$ value. Each row of an index matrix $X$ is generated randomly such that for each row in $X$...

You can see that I have used a kind of template "[some property] of an index matrix $X$ [has these constraints expressed in terms of $X$]" to communicate that the constraints apply to some property of an index matrix that could be any index matrix whilst providing some notation to make it simple to express those constraints in terms of $X$. The problem is that some sentences can start to seem ambiguous. Does the $X$ in "[some property] of an index matrix $X$" refer to the property or the matrix? You can usually work this out with the surrounding context but when explanations get more complex, this becomes trickier to work out. It also seems a bit bloated to keep repeating the phrase "an index matrix $X$".

I have considered other ways of doing this. For example, the template "For any index matrix $X$, [some property has these constraints expressed in terms of $X$]", helps to get rid of the ambiguity, but it seems even more bloated to keep repeating "For any index matrix $X$". Alternatively I could say something like "Let $X$ be any index matrix" and then subsequently use the template "[some property of] $X$ [has these constraints expressed in terms of $X$]" but when the explanations are so long and complicated it can start to read as if $X$ is some specific index matrix and only $X$ has the specified properties and constraints which other index matrices may not have (despite explaining that $X$ is any index matrix in advance).

Are there any writing conventions that people have used to get around this? The main goal I have in mind is to explain things as succinctly and clearly as possible whilst avoiding ambiguity and making it clear that the explanations apply to all instances of the type of object being introduced.

Edit: Just thought I would provide some notes on style and also provide an updated example to fix issues that distract from the main question. In my area of research, dynamical/complex systems or computational models are commonly described/defined in paragraphs (aside from a few equations) rather than providing lots of formal definitions (for example, much writing on random boolean networks is like this). So I hesitate to deviate too much from this style. Here is the aforementioned updated example:

Here, an index matrix is a matrix of positive integers where each row of the matrix is a permutation. To construct an index matrix, the $K$, $N$ and $M$ values of the matrix must be chosen. The $M$ value of an index matrix is the number of columns in the matrix. Each row of an index matrix $X$ contains $M(X)$ distinct integers that have been randomly chosen from the set of the smallest $K(X)$ positive integers, where $K(X)$ is $X$'s $K$ value and $M(X)$ is $X$'s $M$ value. Hence, $M(X)$ must be less than $K(X)$ for any index matrix $X$. The number of rows in an index matrix $X$ is $2^{N(X)}$ where $N(X)$ is $X$'s $N$ value.

Jonathan
  • 459
  • 5
    That description is very hard to follow. Specifically, it's hard to tell which properties you are simply requiring the object to have and which (if any) are properties which follow from other assumptions. I would list the desired properties as bullet points. That is, write something "An $I-$matrix is a matrix of positive integers which satisfies the following assumptions: $\cdots$" Avoid run-on paragraphs wherein it is easy for a reader to get lost. – lulu Apr 25 '24 at 14:33
  • 2
    Also: Examples are always good. Even negative examples, as in "The matrix $M$ is nearly an $I-$matrix but it fails property $iii$" or whatever. – lulu Apr 25 '24 at 14:35
  • 3
    A standard way to define a new type of object in pure maths writing is to put that in a proper Definition. Then you can say "let $X$ be a matrix whose entries are positive integers. Say $X$ is an index matrix if (blah blah blah)". If there are a lot of conditions, then this works very well with lulu's suggestion to use bullet points. A numbered list can also be helpful as then later you can talk about "point $k$ from definition $n$". If there are a lot of properties, you can also break those into separate definitions with their own names. – Izaak van Dongen Apr 25 '24 at 14:41
  • 2
    I know that this is not what you are asking. But. You say that each row of $X$ is a permutation of the smallest $K(X)$ positive integers, that is a permutation of ${1,2,\dots,K(X)}$. So isn't it clear that $X$ has $K(X)$ columns? So where does the "less than or equal to" come into it? I'd find it all easier to understand if you wrote something like "A $2^n\times k$ matrix $X$ is said to be an index matrix if (i) every row is a permutation of ${1,2,\dots,k}$ (ii) whatever (iii) whatever. – ancient mathematician Apr 25 '24 at 15:05
  • 2
    Instead of saying "associated with a $K$ value" and so on, which I find vague-sounding, try something like this: in the definition of "$X$ is an index matrix", one of the defining properties should be "there exists a positive integer $K=K(X)$ such that...". – Greg Martin Apr 25 '24 at 16:13

0 Answers0