well i woke up at 2 pm... ig 8 hour sleep is all i can get lmao. it was so cool to see my friends messages, i mean i have literally the perfect life man, everyones supportive and like most importantly im supportive. i know how good i am and how much i want this lol, and i know this is going to happen like 1000%
anyways, so i replied to my friends messages, and i was talking to this girl and i called her a bop and she got mad and i was like damn, what a fucking waste of time, im trynna be the best ever h0e lmao so i started doing stanley combiiii (from last night). i mean yeah i thought about that problem, like the summation a lil bit more but ig cos i hadnt slept for two days, i just felt so sleepy, so i just came to the drawing room (this place has a special place in my heart man, i remember i used to be up all night reading this math olympiad book on this exact sofa lol) and i put on birds colplay radio and slept (around 5)
i just woke up now, read a lil bit of the cryptography book (by silverman) and obv its like 9:30 now and im bout to head back home, where i will start working again. lets fucking go man, this shit is fun as hell
4:06 am
dang i feel so good, all i gotta say rn is : dont get comfortable at all... wins on top of wins on top and so on... cos youre the best ever. thats what you are. so work and work and work and just have fun, lets go, imma work on combo right now
4:21
i mixed mango monster+ peach monstre+ peach ocean and 6 ice balls and put it in my ishowspeed bottle. degen final boss. also, finally did the eulerian polynomial thingy, it was just a matter of sitting down and focusing lmao
//or something**** [from downloads day2] (lmao i wrote img src=pic1 in my notes instead of attaching the pic here, surely theres a better way to make html blogs)
but wtv, this eulerian thingy, ive alreday done before, so nothing special yet, lets just keep going... birds coldplay radio 4:34
lmao im so fucking smart is funny... but yeah im having fun revising this shit... somethings i revides in the past 5 mins:
strong exceednece, basically consider a permutaiton w (like you can think of w as a funciton or word, your choice) but yeah if w(i)>i its a strong exceedence (SE), if w(i)>=i its a weak exceedence (WE), similarly i defined weak descedance (WD) and SD (i dont even know if descendence is a word, and i dont care, this is not an english class)
but yeah, using the w->w^ bijection, that sends w to its standard cycle form (look it up if youre curious nigga) we see that i index is a SD in w iff i val is a Descent in w^ (descent in permutation means the next element is smaller)... btw this way of thinking of mine, of like you can think of permutation statistics in terms of index or vals is actually so useful now... ok so this shows no. of perms w k WE is equal to no. of permutations w k descents=A(n,k+1)...
trivial if you understand what these words mean, and jibberish if you dont,
but yeah i also found a really cool bijection that sends things w/ t+1 WE to things w/ t SE, basically bijection is v(i)=w(i)+1 (mod n, mod is b/w 1 and n)
so everything that was a WE becomes a SE, and obv val n was a WE and it goes to 1 now, so no shot its a SE now but yeah cool right.
so A(n,k+1) is also no. of perms w/ k+1 Weak exceendecnes...
man wtf am i doing, i spent 8 mins learning ts and 8 mins blogging lmao... fuck this shit, im not a blogger lmao, imma just study shit and then write what i did, dont really need to explain math here , if youre curious about any of ts, dm me, or comment if i have added that feature by the time youre reading this lol
back to work and having fun.... youtube music radio feature is actually goated, man i love yall
5:15 am
well i proved the \sum A_n(x) t^n /n! = \frac(1-x)/(1-xe^{t(1-x)}) result (reproved obv) ... and then i was thinking about like how maj(w) is sum of all indices that are a descent, but what if we did sum by vals on things that are descents, and like sum of vals that are weak exceedences (cos this related to w^ etc) but then idk why i opened whatsapp, and talked to daksh, and saw that the bop girl had deleted some random pics and i was just confused lmao, but anyways, the point of this story is: when youre working, overdo it, like do not stop not for even a single second, push yourself more and more, no complacence man, i really dont care about nothing... get in the flow state and just stay there man, go be the best ever again :)
5:55
i was kinda mad about the picture thing, but then i was like idgaf lmao, im the best ever. but yeah the maj by val thing is an interesting thought experiment, cos obv maj by val statistic is not eqdist. with maj, cos like in indices, the possible values are [1,n-1] but in vals, the possible vals are [2,n]... but like im understanding permutation statistics as a whole better now (like this is crazy, im understanding permutations as a whole better now, ig my theory of the longer you do something the better you get at it, even if you arent working on it is corerct, imagine how much better ill get when I AM working on it all day too)
btw this daily blog thing is inspiring me so much, cos like i dont wanna be a bitch in public lmao.... i gotta get so much shit done yessirski, i wanna be the best ever, and this is so much fun :)
btw yeah sorry, back to the majbyval thingy, i basicaly formalised that sum majbyval (ie. majbyval(w) = \sum _(i is a descent) w(i)) and sumSD (ie sum i is a SD) are equidistributed, like this is obv from the w^ thingy but still its nice to formalise intution,,, i modified the majinv code i had written just to check it also wacthed a couple of reels, conor and ilia are actually so fun, if someone did it, i can do it, and if no one has, then imma be the first... also togi is good. thats the mindset i have man, i already know im the best, i alreayd think of myself as a tech billionaire who has changed the world, i jsut gotta do the plan i have and its going to happen, so i mean yeah just having fun, lets go :)
6:37 am
i reproved the fact that inv and maj are equidistributed, and man hoenstly this was actually brilliant , like i had proved this a couple weeks back, but then i had did mony method, like i just knew the definition of what maj is (again for you guys, its sum of all indices which are descents, like in permutaion 341562 , indices 2,4,5 are descents, so maj here will be 2+4+5=11) i mean i just knew this and i proved they are equidistributed and then found a bijection which gives ts... and honestly, i think the proof i have found may be the simplest possible proof of this result, idk if this has even been discovered before and idc... just the way i discovered it, like writing code making conjectures thinking etc was just the dream life, but agian, this is not a therapy session, we move lmao... ill attach my proof if i feel like it, butyeah if youre reading this, go try this problem out, you know what inv is (no. of inversions in a permuation) and what maj is, so you have the same amount of info as me, try this out and i think youll see why i am so confident in my skills ;)
in short my proof was inv(n+1.k)=inv(n,k)+inv(n,k-1)+....inv(n,k-n) and then i proved the same result for maj and like my main conjecture was no. of permutations with maj=k and ending in n+1-i is maj(n,k-i) which i proved by strong induction :) this gives you the bijection i found too lol
7:41
take me out playing :)
i have revised everything... the bijection, and my notes in which i had a bunch of cool proofs of the \sum c(n,k)a^k =(a)(a+1)...(a+k-1) thingy... but yeah, cool shit :)
again watched a couple reels etc and now back to work... i mean the thing is, im just so motivated to make the most of every single day, like every single blog should be legendary, that i dont even think about tired or nothing, im just not stopping till the day is not legendary, get more done man, btw today i promise was the last day i have wasted time at all lmao... i gotta do tooo much to waste time lmao, lets keep learning, i think imma do apostol ch-2 problems, i mean cos im tired as hell, so like things i have already done and need to revise willbe easier to do since its just grinding lol, lets go :D
8:35 am
just came back from a scooty ride with headphones, beautiful man. its actually crazy how good i feel cos its like nothiing even matters apart from making every day the best story it can. i do need to work even harder tho, so right now, lets do this with no breaks, not even small ones :) apostol ch-2, lets get it (btw apostol obv refers to analytic nt)
8:44
ok this is seriously so cool, i gotta mention this lol... monyatwu.com is now live or archived on the wayback machine!!!! woohooooo im so fucking happy lets gooo.
i mean check it out: monyatwu wayback machine
this is really going to be historic mannnnn woohoooooooo lfg... apostol now :) take me out radio
11:59 am
well one cool thing i did in apostol is, in my notes i had written is f is multiplicative, then f is completely multiplicative iff f inverse = \mu * f (*=times)
so i thought about like what if we drop the f is multiplicative condition, like what if second condition is true , does f have to be multiplicative?
and the answer is no, i found this cool counterexample: f(n) = (-1)^(1+w(n)) where w is the little omega function, the number of distinct prime divisors... btw the wikipedia article on omega function prime omega function was inspirational asl...
i did some more math, man im just exhausted... i ate food etc,,, so i didnt get as much done today as i wanted to, last day fs tho... come on man, no complacence, you got to work... this is your dream right, lets go have fun
you gotta get addicted to the feeling of constant W's