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I'm an amateur 42 years removed from academia so I don't always know "the language".

In a paper I am writing, I specify search limits like $\;4\le m \le 9\;$ and then show, in examples, how some members in the search set yield integers for a function: $\;k=f(m).\quad$ In an example where $f(4)=1\land f(8)=7\;$ I have been using the following notation. $$m\in\{4,8\}\implies k \in\{1,7\} $$

The "sets" can be larger or more complex such as for a cubic where there can be up to $3$ solutions for a given $m$. $$m\in \{7\}\implies k\in\{5,8,3\}\qquad\land\qquad m\in\{8\}\implies k\in\{7\}$$

Does this notation make sense at all and does it make it clear that $4$ corresponds to $1$ and that $8$ corresponds to $7$? Alternatively, is there a better way to express these "summaries"?

This notation is the briefest I could think of outside of showing ordered pairs such as $\quad(7,5)\;(7.8)\;(7,3)\quad (8,7).\quad$ What I want to emphasize is the the "search" found its objective(s) within the specified limits because the point of the bigger "thing" is that what we are looking for can be found $\textbf{without brute force}$.

$\textbf{Update:}$ In response to where this fits in the middle of a paper, here is one of a dozen or more "problems" as I have written them so far. The question remains about the best way to present the results of the search.

$$mk^3-m^3k+D=0\implies$$ \begin{equation} k_0=\sqrt{\frac{4m^2}{3}}\cos\biggl({\biggl(\frac{1}{3}\biggr)\cos^{-1}{\biggl(-\frac{3\sqrt{3}D}{2m^4}\biggr)}\biggr)}\\ \quad k_1=\sqrt{\frac{4m^2}{3}}\cos\biggl({\biggl(\frac{1}{3}\biggr)\cos^{-1}{\biggl(\frac{3\sqrt{3}D}{2m^4}\biggr)}\biggr)}\\ \quad k_2=k_1-k_0 \end{equation}

$$\text{ for }\quad\lfloor\sqrt[4]{2D}\rfloor\approx\sqrt[4]{\frac{8D}{3}} \le m \\ \le \frac{\sqrt[3]{\sqrt{3} \sqrt{27 D^2 - 4} + 9 D}}{\sqrt[3]{2} \sqrt[3]{3^2}} + \frac{\sqrt[3]{\frac{2}{3}}}{\sqrt[3]{\sqrt{3} \sqrt{27 D^2 - 4} + 9 D}}\approx\big\lceil\sqrt[3]{D}\big\rceil$$

$$D=840\implies \lfloor\sqrt[4]{2(840)}\rfloor=6 \le m \le \lceil\sqrt[3]{840}\rceil=10\quad\text {and we find}$$ $$m\in \{7\}\implies k\in\{5,8,3\}\qquad\land\qquad m\in\{8\}\implies k\in\{7\}$$ $$F(7,5)=(24,70,74)\quad F(7,8)=(-15,112,113)\quad F(7,3)=(40,42,58)\\ F(8,7)=(15,112,113)$$

Not related to the original question. The "approximate" limits shown are what I found heuristically before I was able to determine the more complicated "exact" limits. I'm thinking of trashing the latter because the approximate limits are lower and higher and would not add much to the search. Still, the exact limits $are$ exact and omitting them might detract from the precision expected in a paper for publication.

poetasis
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    sets are not ordered, so $m\in{4,8}\implies k \in{1,7} $ does not make it clear that $4$ corresponds to $1$ and $8$ to $7$ – J. W. Tanner Jan 24 '21 at 22:35
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    If it's really just $k=f(m)$, all the info you need is conveyed with $f(4)=1$ and $f(8)=7$ - and that's probably the best way to put a couple values. If you have lots of values, a table of $m$ vs. $f(m)$ could be more appropriate. It seems like your thing is not quite a function - which complicates things (...but a table of $m$ vs. solutions $k$ could still be good if there's enough data to that it's hard to put directly into words). That said, it's kinda context specific - it's hard to be sure what'll sit best in the middle of a paper. – Milo Brandt Jan 24 '21 at 23:27
  • As an aside, $\left\lfloor\sqrt[4]{2(840)}\right\rfloor=6$, not $7$. – Mark S. Jan 25 '21 at 01:04
  • @Mark S. Thanks for the correction. In practice, I've found the ceiling function to be correct so I rounded up instead of the "safer" rounding down. I have corrected the post. – poetasis Jan 25 '21 at 01:16

1 Answers1

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Without further context to override this, a list or table of pairs would be appropriate. Based on the snippet of the text, I also recommend using a lot more words/sentences to break up what's going on. If you read more papers in your area, you'll get a sense for how much they use words to present things clearly.

Note that if there are many similar calculations, extra sentences would get in the way after the first couple times, if it's completely clear from context what is going on. If that's your situation, representing all of the routine steps of the calculation in a table may be appropriate.

So for the snippet in the question, I would recommend sentence(s) along the lines of:

For $D=840$, the bounds for $m$ are $\lfloor\sqrt[4]{2(840)}\rfloor=6$ and $\lceil\sqrt[4]{2(840)}\rceil=10$. The only pairs of integer solutions $(m,k)$ in that range are then $(7,3)$, $(7,5)$, $(7,8)$, and $(8,7)$.

The ordering by increasing $m$ and then $k$ helps the reader to follow along. For more complicated situations, tables may be appropriate. Perhaps something like $$\begin{matrix}m&k\\ 3&4\\ 3&7\\ \\ 4&3 \\ 4&5 \\ 4&12 \\ \\ 5&4 \\ 5&6 \end{matrix}$$or $$\begin{matrix}m&k\\ 3&4\\ 3&7\end{matrix}\,\,\,\,\begin{matrix}m&k\\ 4&3 \\ 4&5 \\ 4&12 \end{matrix}\,\,\,\,\begin{matrix}m&k\\ 5&4 \\ 5&6 \end{matrix}$$

Mark S.
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  • The paper is about a new formula for generating Pythagorean triple. Most of it is $18$-or-more problems to contrast how it and Euclid's formula can be used to find triples by side length, perimeter, area-perimeter ratio, area , or side difference. The "problems" are all presented in the same manner and solved by similar techniques. These abbreviated solutions have already pushed the paper to $15$ pages. Is there a 'brief' form that would be acceptable in presenting search results once the pattern of presentation is shown? I'm thinking the ordered pairs on one line. What do you think? – poetasis Jan 25 '21 at 01:33
  • @poetasis It really depends what changes from "problem" to "problem" and what is known to the reader. Like a table with columns for "equation", "$D$", "$m$", $k$" might be enough if all the calculations are described in enough detail that your audience could reproduce them. – Mark S. Jan 25 '21 at 01:47
  • The odds against there being a new formula for generating Pythagorean triples are enormous. – Gerry Myerson Jan 25 '21 at 02:07
  • @Gerry Myerson I have looked long an hard for prior art using my formula but did not find it, even with a bounty on this question – poetasis Jan 25 '21 at 02:27
  • @ Mark S. These terms D,m,k are all known to the reader long before this problem is presented. I can and do use tables but they generally take up more than one line and, if I \begin{table} \end{table} around an array, it jumps to the head of the next page available. I'm hoping for a 1-line summary of the search results as shown above. – poetasis Jan 25 '21 at 02:36
  • @poetasis if you are not satisfied with something like "We obtain $(3,4)$, $(3,7)$,..." and having trouble getting tables to do what you want, that is a (La)TeX question, not a mathematical one. Questions like "I can't get this table to stay on the page I want" have been asked many times on TeX stackexchange. TeX aside, if you really want to be more compact, at the expense of readability, invent and explain something like "$3:4,7;4:3,5,12;5:4,6$. – Mark S. Jan 25 '21 at 12:06
  • @Mark S I never objected to "We obtain (3,4), (3,7),..." or such and I think I asked if it might be a good way to do it. I do resist the table alternative because it uses so much vertical real estate. Would you care to edit your answer to reflect this suggestion? Up until now I've been tempted to downvote it because it is not useful to me as it is. – poetasis Jan 25 '21 at 12:16
  • @poetasis I'm not sure I understand what you are saying and I think we're talking past each other somehow. To make sure I understand, I'm hearing "please add 'We obtain (3,4), (3,7),..." to your answer". From my perspective, I would say it's already in my answer in the primary suggestion of "The only pairs...are then $(7,3)$, $(7,5)$...". The original question seemed to be about notation, and not best wording. You can certainly use more or fewer explanatory words in the sentence, as appropriate for the context of what you're writing. It's quite possible I'm still missing what you're saying. – Mark S. Jan 25 '21 at 12:26
  • @Mark S. I'm saying delete everything except the abbreviated form. Instead of " The only pairs of integer solutions $(m,k)$ in that range are then $(7,3)$, $(7,5)$, $(7,8)$, and $(8,7)$." use "We obtain $(3,4), (3,7)\cdots$" if this is a valid solution and acceptable to a potential publisher. It could be worth 25 pts. – poetasis Jan 25 '21 at 12:31
  • @Mark S Maybe I just hate doing in 3-to-10 lines what can be done in one, and which can be understood at a glance without, for instance, looking to see what are the column and row headings of a table. I didn't mean to sound unappreciative or demanding. I'm just doing with little sleep right now and I would like to give you credit (points) for the shorthand version of ordered pairs you suggested if you indeed think the $1$-line approach is acceptable for publication. The reader will already be familiar with the format because I show examples after every solution. – poetasis Jan 25 '21 at 12:57
  • @Mark S. I've decided to upvote you an check your answer correct because you did provide the support the idea of ordered pairs in abbreviated form in a comment. I think I have what I need and, now, after having gotten more sleep, I can see that and I thank you. – poetasis Jan 25 '21 at 18:58