Category Archives: collatz

collatz mod3

ok, this is a sort of shocking discovery at the new year. it is almost obvious to look at properties of iterates wrt remainder by division. there was a years-ago look into factorizing. also there are years-old looks at the bit (base2) diagrams, and this was adapted to do base3 diagrams, but didnt notice anything unusual and left them unwritten-up. did not notice this until just now, but it might have been in those visual diagrams. was looking at some older glides associated with mix30d. was curious about the mod3 behavior intra glide pre/ post peak, actually wondering about existence of repeated 3 factors. was quite surprised to find this major discrepancy/ differentiator. randomly sampling/ spot checking, these glide iterates were never divisible by 3! also, for the other two mod3 values, the distributions are different over the iterates pre and post peak. for climb the mod3=2 case is about 3x the mod3=1 case. for descent the ratio is closer to about 1.5. this is demonstrated in this simple code.

three7.rb

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collatz finetuning

the last idea from 10/2018 was bouncing around in my thinking some, it left open a question about “degree of monotonicity”. it shows that the max 0/1 runs seems to be almost monotonically decreasing even in a climb/ glide. havent really looked at the monotonicity of the max 0/1 runs all that directly with the nonmono run length measurement. ofc all the recent analysis is related to monotonicity, but not exactly tied to nonmono run lengths. there is some relation. at this point am interested in finding long nonmono run lengths in the max 0/1 runs sequence. last months genetic algorithm code looked into this but for a fixed (“1d”) bit width. what would that look like for variable bit width (2d)?

the last “2d” genetic algorithm search code was mix32 from 9/2018. the mix series is aptly named because it mixes quite a few ideas across many algorithms. it has single dimensional genetic algorithms, bitwise optimizations, and 2d genetic algorithms (ie both within given bit widths and over multiple bit widths). am going to rename the 2d genetic algorithm here hybrid.

got all kinds of cool ideas for refactoring/ streamlining working on it. it has some abstraction for all the initialization and combination/ crossover operators. it has a very sophisticated 2d algorithm that is similar but different from prior algorithms. it has no restrictions on expanding “bit bins” except that the expansion operator does it only by a max of 3 bits. it dynamically analyzes bit bins for fitness similarly to analyzing fitness within the bins based on top performing candidate in each bin, and combination operators work on the top performing bins. it currently throws out the most underperforming bit bin and currently keeps the total bit bins to 50 and the size of each to 50 for max 2500 candidates. the code enforces a minimum of the max 0/1 runs in each iterate, here 6. starting bit size 10. the recent initw bit string initializer code turned out to be crucial after the minimum was added because the other initializers dont seem to create many iterates with sufficiently large max runs.

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collatz deja vu

❤ 💡 didnt expect to focus on collatz so much when starting this blog, but hard problems require (sometimes intense) focus! as you can see/ tell it has sucked me in. in the sense of “femme fatale” maybe? such a much more interesting/ evocative metaphor imho than others that are proposed.

am gearing up for a new saga/ installment/ pivot (because last mos saga ofc wasnt quite enough, lol!). it requires a bit of “psyching up” as the vernacular goes. it involves dredging up some (now) very old ideas that are potentially quite powerful but at this point have cobwebs on them. blast from the past™/ back from the dead™ one might say dramatically. yeah its a possibly big success story of long having something valuable/ powerful in the back pocket and resuscitating it. its also a case of years-long near miss/ strong/ shiny/ sparkling lead that came close at the time but didnt realize exactly/ completely pursue it in the moment. what goes around comes around™.

background (“static”), have been assigned a new fat/ significant prj at work and am a bit bummed/ chagrined/ annoyed about that because it cuts into this “fun time with collatz” (so to speak!). usually end-of-year period allows some respite/ hiatus as the code goes into a temporary freeze period. whatever

😎 more painting scenic/ colorful background: on the porch in backyard this second with cap + sunglasses + headband + tshirt + long sleeve shirt + sweatshirt + thick black leather jacket + long underwear (“layers”) vs the sun rise, next to nice winebarrel firepit, rose incense in the shamanic wood tower holder imported from eucador, admiring my nice sculpture collection (from fish tank decor, repurposed!) including the 2 new 2ft faux rock dragon protectors doing a great job so far. if someone is really nice/ or maybe just human to me in comments, may post picture highlights! believe me its well worth it! it started out at 40°F on this fall day but already up to 50° in about only 30m. the headband + bike gloves with open fingertips defn help! am lately thinking/ musing that maybe the problem difficulty should be measured not in midnight oil mentioned earlier but in incense sticks burned, and it is quite numerous/ sizeable so far!

other news: am starting Frenkels book Love and Math 2013, was tipped on this half a year by colorful chat denizen 0celo7 RIP (its a long story™… oops post correction it was cohort Balarka) who pointed out the extraordinary porn star “erotica angle” lol! wow, iirc skipped buying this at the time, thought it was a little too fluffy maybe, but now think its an instant classic. he talks about beauty in math a lot the way Hardy did in Mathematicians apology. poetic! what a great writer!

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collatz fusion

❗ ⭐ ❤ 😎 😀 hi all. it was a very lively week last week it would seem "all cylinders were firing". the mini adventure crosscut other blogs, reddit/ social media, multiple stackexchange sites and chat rooms, mainstream media, top world mathematicians eg longtime favorite heroes eg Tao/ Gowers, and a world class award/ prize venue ie Heidelberg Laureate forum— (eg what (personal?) connection the skeptic naysayers ask?— yes! eg did get substantial hits from their blog referrals!). it is a relatively rare event for even several of those “different spheres” to intersect.

(looking on the bright side!) the Atiyah announcement ties in with some key interests/ themes of this blog covered in the past such as attacking top open problems, (cyber-) collaboration/ peer review, scientific psychology/ culture, tracking the award/ prize-winning leaders of the field, tracking current leading edge research/ controversy which sometimes also leads to real breakthroughs, empirical/ computational/ coding approaches to math theory research, physics/ math research connections/ overlap, number theory, etc., did have some real fun! it seemed to have paid off to jump in the fray with a big analysis that got many hits and a lot of comment feedback, the former uncommon and the latter a rarity for this blog. emotions ran high! the story was not the happy/ breakthru/ historic ending hoped for but it was nevertheless quite the story, and definitely still historic!

as stated in chat, do think instead of a “dud”, alas, a “Big Kahuna™ (Hooked)” somewhat on the level of Perelman or Wiles proof is on the intermediate horizon and will be seen in our lifetime. P vs NP, Riemann, Collatz, not sure which, they seem to come along about once per generation; anyway from this episode looks like all the troops are lined up and ready to pounce so to speak. (anticipating objections…)

some might laugh at putting Collatz on that list, but it really is that important to me at least, and suspect it could rise in wider community attention/ importance with an ingenious solution, esp if its a potentially broad new technique that could seemingly have applications to other problems, say algorithmic/ computational/ deep learning/ auto thm proving aspects— ie exactly what is long sought/ advocated/ promoted here, and have believed for decades something like that is lurking in the deep so to speak. for those students of history, there is no insurance that a problem thought significant will stay that way/ aka “stand the test of time” but likewise, sometimes seemingly inconsequential problems rise in importance. theres an element of fashion to science/ even math, but in a deeper sense… maybe more like uncovering hidden gold veins, possibly bulging deeper

meantime have also been banging on some interesting angles with the collatz research, and there was some cross pollination as commenters reacted to my referencing Tao-Polymaths new “number crunching” Riemann attack. from long observation it seems some (minority of) mathematicans are “triggered” by CS-oriented approaches to math problems, and they have my full sympathies, but some are also not very conversant with or tend to pushback against long history of world class breakthrus in the area, covered in some other blogs here/ many writings elsewhere. everyone is entitled to their own opinions, but not their own facts™…

as promised/ foreshadowed/ alluded last month, did want to share/ unveil/ unleash my shiny new analogy/ “elevator pitch”, now established as a nearly monthly tradition around here. have been tracking physics research/ breakthrus in this blog since its inception. one lively area is fusion research. its very broad to try to look into, but basically there are new directions/ developments being pursued worldwide in this area and one hears of some advances. in particular “stellarators” are gaining some viability. some scattered refs in the blog, am collecting links for a new one.

in fusion research the fiendishy difficult problem is trying to control a levitating extremely-high-energy plasma using magnetic fields. in this way there is some similarity to atom trapping technology that advanced rather extraordinarily in our lifetimes, in the last few decades. however atom trapping, as scifi/ hitech as it is, seems to be relatively “easy” compared to fusion control.

in fusion research in contrast the plasma is highly unstable, much more than an atom is, the atom despite its internal energy fluctations is nearly stable in comparison. its looking like extremely sophisticated control technology is going to be required in stellarator fusion. it looks like it will require extremely fast computations to devise a control system that reacts and counteracts the very unstable/ dynamic “modulations” in the plasma. one is reminded of the classic Lorenz attractor problem which again is highly stable in comparison. lately there are new insights and maybe something like the worlds most sophisticated/ fast responding AI algorithms responding to complex measurements within milliseconds, and some very advanced theory may be invoked. another emerging beautiful combination/ fusion of physics + TCS… stay tuned more later…

anyway my new analogy with collatz is as follows: collatz trajectories are sort of wild, unstable, and even “fractal” like in the sense of a plasma. to solve the problem, one has to somehow “trap the plasma”. the plasma escapes different control methods, this is the sense in which some parameters measuring it become unbounded. others somehow bound it even as the plasma gets larger/ more energetic. the plasma is chaotic yet somehow there may be an (extraordinary, to say the least) orderly control method that converts disorder into order (down-regulates the entropy so to speak). these ideas were recently carried out in the sense of the nonmono run lengths of the right side of glides apparently looking bounded compared to left sides (despite “push-up” optimization applied on both, ie cf bitwise8 experiment) and there are many other experiments with this theme eg graphs where some lines/ measurements trend up with increasing trajectory lengths as others flatline/ plateau.

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collatz search

the word “search” is something to ponder wrt this effort in math “re-search“. there are different optimization algorithms as a theme but in general theyre different types of searches for trajectories. in a way its all about searching for some kind of invariant property of trajectories, both with the computer/ algorithm and more generally in the research (“program”). have come up with many ingenious search methods and that effort continues.

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