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sniffnoyWhat with the conclusion of writing this first paper (gods willing) -- and the end of classes -- I have some brief time to actually do some *thinking* about integer complexity again before Jeff makes me get started on writing the next one[0]. (I believe much of what I am thinking about has already been solved by Juan Arias de Reyna, but I will read his solutions later; first I want to think on this myself some more.) Sorry, no details right now -- well, unless you think you can infer them from the rest of this post!
In earlier drafts (that included material we definitely did not have time to cover in this first paper) I defined a class of ternary families I called "primary ternary families". I am thinking now this was not such a good name. (Also, people keep telling me "ternary family" is a terrible name, and I assume they're probably right, but since right now I'm just talking about naming things for the purposes of my own thinking, I'm going to ignore that right now.)
Rather I am thinking that would be a better name for a slightly more general class of ternary families. (The old meaning is still important, but slightly less so.) But I've used this name long enough that I would have trouble switching it over in my mind. And perhaps a more suggestive term could be used, anyway.
The intutition is that these "primary" ternary families are somehow bulky or substantial. And with the more restricted meaning, they're even more bulky. I like the idea of calling "bulky" what previously I called "primary". But what about the more general class? "Substantial ternary family" is a bit of a mouthful. "Meaty?" No, that's a little too out there. "Solid?" Unfortunately, we ended up using "solid number" to refer to additively irreducible numbers, so that's out. Though maybe we can still change that back to the term we were using previously -- we called such numbers "chunks". I still like this better. Actually, so did Josh, but Jeff had problems with it and we decided it was more important to get the thing done than to bikeshed about the term. But I suspect Jeff slightly misunderstood how we intended to use the term, so perhaps if that is made clear he will change his mind.
Yeah, I think I'll stick with "substantial". (Anyway, these things might well end up as polynomials, and "substantial polynomial" is not so bad to say.)
-Harry
[0]Also some time to do some more coding on the classifier, which is most of what I've been doing. The core of it has been done for a long time, of course, but it still needs to be changed to use exact arithmetic, and I'd also like to add some additional output capabilities. Like maybe, "enter a real number (in a format I can work with), get back the order type of defects up to that real number". I don't know if I'll actually implement that one. Computing order type is straightforward from the existing core tools, of course; it's the reading in a real number in a potentially quite general format I find daunting. :P But I guess I can always just implement the actual function itself, and then say "open the program up with ghci if you want to use these additional features"...
In earlier drafts (that included material we definitely did not have time to cover in this first paper) I defined a class of ternary families I called "primary ternary families". I am thinking now this was not such a good name. (Also, people keep telling me "ternary family" is a terrible name, and I assume they're probably right, but since right now I'm just talking about naming things for the purposes of my own thinking, I'm going to ignore that right now.)
Rather I am thinking that would be a better name for a slightly more general class of ternary families. (The old meaning is still important, but slightly less so.) But I've used this name long enough that I would have trouble switching it over in my mind. And perhaps a more suggestive term could be used, anyway.
The intutition is that these "primary" ternary families are somehow bulky or substantial. And with the more restricted meaning, they're even more bulky. I like the idea of calling "bulky" what previously I called "primary". But what about the more general class? "Substantial ternary family" is a bit of a mouthful. "Meaty?" No, that's a little too out there. "Solid?" Unfortunately, we ended up using "solid number" to refer to additively irreducible numbers, so that's out. Though maybe we can still change that back to the term we were using previously -- we called such numbers "chunks". I still like this better. Actually, so did Josh, but Jeff had problems with it and we decided it was more important to get the thing done than to bikeshed about the term. But I suspect Jeff slightly misunderstood how we intended to use the term, so perhaps if that is made clear he will change his mind.
Yeah, I think I'll stick with "substantial". (Anyway, these things might well end up as polynomials, and "substantial polynomial" is not so bad to say.)
-Harry
[0]Also some time to do some more coding on the classifier, which is most of what I've been doing. The core of it has been done for a long time, of course, but it still needs to be changed to use exact arithmetic, and I'd also like to add some additional output capabilities. Like maybe, "enter a real number (in a format I can work with), get back the order type of defects up to that real number". I don't know if I'll actually implement that one. Computing order type is straightforward from the existing core tools, of course; it's the reading in a real number in a potentially quite general format I find daunting. :P But I guess I can always just implement the actual function itself, and then say "open the program up with ghci if you want to use these additional features"...
