Category Archives: collatz

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 rescuscitating 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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collatz marathon

yep, its definitely a marathon at this point, maybe now mostly comprised of month-long sprints (applying “agile” philosophy to mathematics/ research in the weird hybrid style typical of this idiosyncratic approach… thats not a bug its a feature™… or maybe in some sense that the historic Thai cave rescue was also marathon…). heres another new idea building on prior ones. earlier analysis looked at the “sibling trajectory” and found a strong correspondence with the trajectory. a fairly basic idea is to compare how much of the sibling trajectory matches wrt the climb length of the glide. in a sense the sibling trajectory “covers” some percent of the climb length.

using the latest generated glides (mix26 scheme) this finds that in ~40% of the cases the sibling trajectory covers the full climb. in other words in 40% of the cases the computation to find the highest point of the glide is exactly repeated in a “lower” glide/ trajectory. in the remaining “uncovered” cases theres still 1.0-0.3 = ~70% coverage at worst and more typically/ on avg in the ~95% range. the graph is the ‘icm’ variable in impulse format which is matching length of trajectory and sibling divided by ‘cm’ glide climb minus 1, here 0 means matching the exact length. as seen in the graph the non-fully matching siblings still cover most of the climbs.

review84.rb

review84

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collatz new strategy

hi all. on vac this week & doing some new stuff (happy BTD US). there is a semifamous thousands-year old quote by sun-tzu maybe not yet contained in this blog (its been going thru my mind for quite awhile now, but wasnt able to find it in the blog via google). it is a quite favorite quote of business consultants which might tell you something about modern “leave no prisoners” business attitudes/ culture in our at-times militaristic/ hypercapitalistic modern age. (dramatic alpha-male stuff, but to put it more bluntly, one with a conscience/ empathy/ independent mind might wonder about the “fine print,” ie how many “enemy…” men did sun-tzu kill personally or oversee killing as a general? …or even humans which includes women/ children? oh but ofc its utterly metaphorical right?) 😮 😳 o_O

Strategy without tactics is the slowest route to victory. Tactics without strategy is the noise before defeat. –Sun Tzu.

collatz has been described as the very impenetrable/ unconquerable adversary. strategy/ tactics both play a key role and have commented at length on both. they are like a yin-yang combination. victory will likely not come without some kind of balance between the two.

have been more/ very tactical for quite awhile but have been musing on some overarching strategy/ perspective/ pov lately, thinking it all over at current point. this involves more abstraction.

couldnt find this basic idea pointed out in old blogs. was it? the key question is to prove f(x) < g(x) for all x. here f(x) is collatz stopping distance or some similar metric and g(x) is “any recursive function” (either time/ space bounded). now apparently f(x) in many related forms has extreme entropy, the “needle in haystack” property, and also “fat/ long tails” distribution, and fractal. earlier blogs have outlined the idea that it appears that victory seems to lie on the path of decreasing or minimizing entropy somehow. g(x) can be regarded as an orderly function from analytic mathematics and f(x) is “far from it” in the sense of being extremely disorderly.

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