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Hi! It’s Monday, my name is Colin and I am a mathematician. What can I help you with this week? Bonus points if it involves continued fractions.

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@michaeljo94 I believe they were first used in India about 1500 years ago!

@michaeljo94 The Collatz conjecture? My understanding is that that's a long way from being solved.

@michaeljo94 I'm sorry, I'm trying to figure out if you want help with something.

@icecolbeveridge what is a sheaf (in math)? a friend tried explaining them to me a couple of years ago and I couldn't understand anything. I've had one semester each of basic category theory and real analysis.

@danielhglus Alas, that's not an area I know anything about, and a few minutes' reading hasn't enlightened me :-(

@danielhglus My doctorate was about the Sun's atmosphere -- a mix of geometry, numerical analysis, basic topology, vector calculus and counting. Since then, I've mainly been teaching post-16 and doing recreational maths. It's much more fun 🙂

@subleq I'm fairly sure Riemann is true. Whether his hypothesis is true... I'm not telling you :-P

@icecolbeveridge Are all continued fractions solvable (in the sense that one can easily determine if they converge to a value, and if so what they converge to)

@icecolbeveridge Are you only consider CFs with numerator 1? I think they all converge, so you must be thinking about "non-standard" CFs ... yes?

(Drive-by tooting ... I have a busy day. I wonder if I could interest you in an idea of mine).

CC: @loke

@ColinTheMathmo Always happy to consider your ideas (especially if I can fit them around childcare commitments...) 🙂

@icecolbeveridge Here's one presentation of the context and initial ideas:

solipsys.co.uk/cgi-bin/DiscDAG

I have a "more traditional" web page, but I'd also be interested in your reaction to following the ideas on the diagram.

(My attention will be patchy, as, no doubt, will yours.)

@icecolbeveridge If you're interested in contributing to that discussion I can send you a magic link. That's the setup ... the actual idea comes next.

@ColinTheMathmo I've bookmarked it to read later on (there's a likelihood of being interrupted by squabbles currently, which makes concentrated thought difficult.) I suspect it's not quite in my wheelhouse, but I'm happy to look and see if I have anything intelligent to offer 🙂

@icecolbeveridge Noted ... when you can the chance, and as always, no obligation even to read it, let alone spend time.

@loke Hm, good question. I think simple CFs always converge to real numbers -- but the conditions for generalised CFs to converge are not fully understood ( en.wikipedia.org/wiki/Converge )

@icecolbeveridge Right, as suspected you're dealing with the generalised case. That's a rabbit hole down which I dare not venture. Just now I can't afford the time to be nerd-snup again.

Looks to be an interesting question, though. Good luck ...

CC: @loke

@ColinTheMathmo (I am on much firmer ground with simple CFs, but I think @loke's question was about generalised ones.)

@icecolbeveridge @ColinTheMathmo Yes, it was about generalised ones. 🙂 The simple ones "seems" to always converge, since the denominator is always decreasing, no?

@loke Yeah, I'm pretty sure simple ones converge (I'm currently at the "handwavy" stage of sureness rather than "I feel like I could convince someone" -- once I give it some thought, I expect it'll drop out quite nicely.)

@icecolbeveridge @loke in general I tend to prefer the Śleszyński-Pringsheim or van Vleck theorems (cf. dlmf.nist.gov/1.12.v) for proving convergence, since their criteria are often easy to verify. On a terminological note, I tend to use "continued fraction" in the general sense, and use the specific term "simple continued fraction" if all the partial numerators are unity. (cc: @ColinTheMathmo )

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