These guys have floodwater jurisdiction; capitalists would be motivated to re-route it and to sell it to places that need it like, quickly, but could also over-sell and damage river ecosystems if not very carefully regulated. I do think the Colorado-serviced region should pay significantly more for its water, even if said water is stabilized; you don't want to motivate population booms that re-destabilize that water, by not making it do so, among a few other more nitpicky points I can consider regarding fairness of draining a river and having it recovered for your a$$. And the money should go back to the responsible civilians along the Mississippi and the Missouri who should havebthe water rights. Idk. Hopefully they can all figure it out. That being said, the labor will be outsourced, and I think we have a natural knack for it in our physiques, but need better safety training and maybe etc. for construction workers. Those safety stats are terrible. Or maybe it will just snow forever in the Rockies, out-doing that 10.3% snowpack recovery loss from thermal change effects minus greenery change effects calculated in that recent paper, and we'll never need to worry again. Or we could let the Colorado-serviced region dry up and displace 40 million people. All good options.
Elleard Felix Webster Heffern’s Post
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Okay, now it's clean, checked in Excel, works for all congruencies of the degree =). #Chebyshev #MarkovNumbers #Unicity #UnicityisComing #webbiestar As noted before, this is to ensure the degree on the polynomial is low enough relative to the variable, to approximate it simply with the variable raised to the relevant power =).
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#Chebyshev #MarkovNumbers #Unicity Wow. So in my complete and utter misery realizing that I'm basically well... let's not go into my intelligence level at the moment, given that I literally derived Chebyshev polynomials in the past... I, baffled over an error, decided to see how something played out with like, actual numbers. Aaaand we have a new identity that is just, crazy powerful. I was /so/ right to go with b being a power of 2, because then you can just recursively bust down those U_b-1s, and U_n-bs if you had an odd n... what is the best way to optimize this thing?! This_is_awesome.
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#MarkovNumbers #Unicity Formula is testing well now, but arguably "not good enough," at least when I use the approximation at the bottom. I want to find something that approximates super super well, well enough to stay within bounds and not touch other Markov Numbers' territory. Obviously I could just expand out the whole expression, but that is very clunky, especially if you're doing it ad absurdum at the end of the proof. ANYWAY, I do need to /test/ these polynomial expressions. I've tested the approximation for approximation, but not the derivation itself for perfection. It is soooo late though. I need to eat and take a nap with Miss Kelly. DW: when I said I had it today, I meant it, in that I "knew what to do." That being cleanly approximating the polynomials like this and then layering through specifically that ln(Un(ym/2)) term which pretty much on its own declares Unicity. Something's bugging me about the present derivation but it's too late to figure out what, now. We're at the "nitty gritty" stages now. I'm probably just foggy from doing this for 12 hours straight and not seeing something staring in my face either to clear up or to clarify how to fix my subconscious concern. Like the final product of q_j values: that's actually going to be /smaller/, I think, for something like 37666, when 37666 actually should have the highest ln. I'm just itchy and groggy and will have to run the total calculations to see where my full pre-approximated formulae stand, tomorrow. Perfection or bust =).
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#MarkovNumbers #Unicity THAT took all night to edit. Those terms pulled out at the beginning were definitely not supposed to be Chebyshev polynomials =). I actually figured this one out a while back, glad it came up =). Complex functions...
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Okay now it's ready to test I think. I should really stop calling them "triangular numbers of varying order;" that's not as well understood by the Math community, referencing Pascal's triangle by the -60 degree diagonals rather than the rows. But you know? That's how I recognize them. And that's how they emerge in this case, so... #MarkovNumbers #Unicity
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I swear I'm not insane... these effects are important to me because I'm a gay person 😝. #MarkovNumbers #Unicity #procrastinatingthetest...
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Well that is ludicrous and fantastic if true... the deduction as stands suggests that if you're looking just at "n" values that are multiples of 4, each time you pass another power of 2, the Chebyshev Polynomial of the Second Kind gauges up the effective power to which it raises x, treating the most recent passage as a double power and everything past that as a single power. #MarkovNumbers #Unicity
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#MarkovNumbers #Unicity Good God... I think I need a break =). Like every single one of these formulae is gorgeous but I need to take my eyes off to say, "THAT's the one I want with THAT one to give me a really clean ln value with not too much of a remainder." Honestly, my first "go" was great, but it didn't address the super-high-degree polynomial situation, which is a very important thing that needs to be addressed. This version simultaneously addresses all situations, but that formula to my eyes does not look like, amenable to me taking its natural logarithm any time soon =). Maybe if I just wrote U_b(x/2) and replaced those terms for now, it might evidence more clarity to me...
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#MarkovNumbers #Unicity Now this is feeling more like it, but that formula for U_b (x/2), where b = 2^a for some maximal a < ln_2 (n) -- that one's a littttle intense. I think I can write something that looks more like a product than a sum, there.
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Now THAT is a sexy-looking expression (but I can't promise it's perfect yet). I did a really smart expansion with the polynomials this time that allowed me to take a natural logarithm VERY easily. #MarkovNumbers #Unicity
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Let's get Gaussian. Did you want the triangle fries, the electric sandwich, or the normal platter?
1yOh, also, if it can be done, don't charge native peoples extra for the water, only non-native peoples. Like, come on. They shouldn't even be amongst those forced to make cuts.