Mathematics - 2022-05-23 (page 2 of 2)

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5:01 PM

Given $n$ strings of length $t$: $s_1,\dots,s_n$ find $l\leq r$ that minimizes $r-l$ with the condition that for any $i\neq j$ there is some $x$ with $l\leq x \leq r$ such that the $x'$th character of $s_i$ differs from the $x$'th character of $s_j$.

Can you define $l$ and $r$ for me please?

Also $i$ and $j$

Given $n$ strings of length $t$: $s_1,\dots,s_n$ find $1\leq l\leq r \leq t$ that minimizes $r-l$ with the condition that for any $1 \leq i\neq j \leq n$ there is some $x$ with $l\leq x \leq r$ such that the $x'$th character of $s_i$ differs from the $x$'th character of $s_j$.

integers

all 4 of those are integers

Oh, I see now. Thanks. Lemme read that until I get it proper.

I'll be honest, I'm still struggling to read what you wrote, Asinomas, but from what I get out of that so far, it seems to be slightly more specific than my particular problem. Different character? Perhaps, but in a base with only two characters to choose from, that is already exhausted by having more than two values.

What's the standard way to measure the amount of information contained in a set with bits as a metric? I've seen 3b1b use that before in his videos on wordle.

5:20 PM

sorry I don't know

Or like how they get the number $\log_2(e)$ for the minimum number of bits per key that a general perfect hashing function requires for optimal space.

I saw errichto was saying he didn't agree with that wordle vid by 3b1b

Well he released a new one to address the issues of the first one in which he claimed that crane was the probabilistically optimal first choice for wordle.

oh

Not really relevant though to what I'm asking :)

5:22 PM

what is the optimal strategy for wordle

if the only available wordle is mexico

Can't remember what the new word is. youtu.be/fRed0Xmc2Wg

soare apparently

I have only played wordle 3 times but succeeded all 3

is that good or w/e ?

If $X$ is an infinite set, then the finitary alternating group on $X$ can be defined in the following equivalent ways: 1. the group of all even permutations on $X$ under composition; 2. the kernel of the sign homomorphism on the finatary symmetric group on $X$.
First, what does even permutation mean if $X$ is infinite? Second, how does one construct the homomorphism mentioned in statement 2.?

Prithu biswas leftmse

@Asinomás yeah it is good.

pog

5:34 PM

user i agree the question is missing context. even in the finite case, the definition of even permutation is sometimes made in terms of the sign homomorphism.

i assume the finitary symmetric group on X is the group generated by permutations that fix all but a finite number of things.

those permutations do have a well-defined sign. but without knowing how 'even' is defined in the infinitary case it is not clear how the statement is not a tautology.

4

Q: Alternating group on infinite sets

It is well known that the only normal subgroup of $S_n$ is $A_n$ when $n\geqslant 5$, and that $A_n$ is also simple. Furthermore, $A_{\infty}$, the even permutations on $\mathbb{N}$, is also simple. This lead me to wonder about the following: Take a general set $X$ with cardinality $\kappa>\alep...

group-theory permutations

@leslietownes Yes, that's what I take "finitary symmetric group" to mean---those permutations with finite support.

that sounds like a dope approach

i see that "generated by" above can be removed.

i could imagine defining things so that the statement is a tautology. is it from a textbook?

as opposed to a non-text book?

or a non-book ?

5:48 PM

yeah. for example, a web page.

a friend.

graffiti in the bathroom.

graffiti not in the bathroom.

in the non-bath room

or in the non-room

i have a sign ready for any political protest. it says: FREE ABELIAN GROUPS

haha that's a good one

only one free group is abelian :(

but there are many free abelian groups

I guess it needs parenthesis

or something

the category people have a language for this.

where you do the 'free' bit. they love stuff like this.

I don't understand this $$f(x)=\sum_{p^v \le x} \frac{1}{v}$$

I would understand if the lower bound said $v \le x$

but when the extra $p$ (primes) is added I don't get it

6:05 PM

you can do research in category theory?

@leslietownes No, it doesn't come from a textbook. I got it from this page: groupprops.subwiki.org/wiki/Finitary_alternating_group

The context is that of Houghton groups.

Given $n \in \Bbb{N}$, then Houghton group $H_n$ is the group of permutations that are "eventually translations" on the set $\{1,...,n\} \times \Bbb{N}$. More precisely, a permutation $\sigma$ on $\{1,...,n\} \times \Bbb{N}$ is in $H_n$ if and only if there exists integers $N \ge 0$ and $t_i = t_i(\sigma)$ for $i=,...,n$ such that $\sigma (i,k) = (i, k + t_i)$ for all $k \ge N$.

Apparently the second commutator subgroup $H_n''$ is the finitary alternating group on $\{1,...,n\} \times \Bbb{N}$ and I am just trying to better understand this.

6:49 PM

Linear map from V to W is completely determined by its values on basis of V. Then why is the following a violation: Suppose U is a proper subspace of V and let S be in L(U,W) then a basis of U i.e. $u_1,u_2,...,u_k$ can be extended to a basis of V say $u_1,...,u_k, v_{k+1},...,v_n$.

Now, if I define $T$ on V as: $Tu_i =Su_i$ and $Tv_i=0$ for every $i$, then clearly T is not a LT.

(also assume that S is not zero).

There is some u such that $Su\ne 0$ so $T(u+v_1)=0\implies T(u)=0,$ which contradicts the assumption that S is non zero.

But I don't understand why T fails to give a LT.

If you're defining $T$ in this manner, then it's definitely linear. What would *fail* to be a linear transformation is

$$T(v) = \begin{cases}Sv & \text{if }v \in U;\\0 & \text{otherwise.}\end{cases}$$

Because then, if $u \in U$ and $v \in V \setminus U$, you'd generically have $T(u + v) = 0$, but $T(u) + T(v) = T(u) \neq 0$.

Speaking of category theory terminology, ncatlab.org/nlab/show/red+herring+principle

category theory is a waste of time

The way I have defined also should give: T is not a linear transformation. I think that this is so by taking v as any of v_i's.

But, defined in this way on basis elements, $T$ succeeds in being linear---$T(u + v)$ is no longer forced to be zero by definition, whereas it is in my case.

7:02 PM

Oh right.

Or said differently, the basiswise definition only makes $T$ zero on a subspace, but my definition makes $T$ zero almost everywhere.

$u+v_i$ is not in the set {$v_{k+1},...,v_n$}

Indeed.

@geocalc33 My guess is that you find the highest $v$ such that $p^v\le x$, for all primes $p\le x$. Then sum the reciprocals of all those $v$'s, one for each $p$

Alternatively, you need to include each $v$ for each $p$. Eg, if $2^4\le x$, then you need to include each of $(1,2,3,4)$ for $p=2$

7:43 PM

anyone familiar with the gram series?

sounds like something a rapper would boast about

my gram series is divergent

[this means that i deal a lot of cocaine]

so is mine

8:02 PM

@geocalc33 Not really. It looks like it gets slow for large x, and according to mathworld.wolfram.com/GramSeries.html it has been proven to be worse [than Li(x)] infinitely often by Littlewood. There's only 1 question about it on the main site, which I guess you've seen. math.stackexchange.com/q/1506980/207316

8:13 PM

If it's not converging, you probably need to use higher precision arithmetic. One of the comments on that question mentions using 100 digits.

8:27 PM

@PM2Ring I came up with a series that's in between the Gram series and Li(x). I think I can ask whether it's worse than Li(x) infinitely often

@PM2Ring worse than Li(x) at measuring primes?

8:48 PM

@geocalc33 Yes, worse than Li(x) at counting primes.

9:24 PM

Why doesn't anyone visit meta?

I have made mental notes that meta sucks and I shouldn't go there

I've visited and posted a few times in 9 years.

but I still show up in the front page by participation

It's mostly CURED people and "haters" from other SE sites.

who post there

I could be an excellent hater.

I am an excellent hater

But I do it in silence

most of the time

1 hour later…

10:40 PM

Oh dope the last user to post anti hater stuff on meta got suspended

Gotta hate haters’ haters.

I would argue it's not compulsory

You mean you might hate the haters of haters’ haters?

I feel like you might have added an extra h word

but sure

I thought I had it right :D

10:51 PM

you probably did

Wait, I thought SE was just a chat network, you're telling me there's a whole site here? With posts?

What is a site? Can you cite me a sighting?

well the SE chats are split into three I think

I don't remember well

I used to do a trick where I would swap my chat handle from one account to another

but then some mod decided to break my trick

I think SO chat and meta SE chats are separate

and then there is all other sites

A plate of momos

whats up gamers

nice vid

A plate of momos

11:02 PM

thx

which anti hater stuff r u talking about

a lot of riots have been taking place on meta as of late

mad world

I was talking about the 50k user that got suspended for a year

A plate of momos

O

I feel like with the current state of MSE, this is the most fitting song for the people arguing on meta

Tears For Fears - Everybody Wants To Rule The World (Official Music Video)

I disagree

Actually no one wants to rule mse

A plate of momos

what do you propose is the most fitting then?

except for losers

A plate of momos

11:03 PM

oh

oops

the song is good

A plate of momos

I guess I mean rule in a weaker sense like

you know whose opinion gets applied

I don't think so

A plate of momos

why so? Some people want EOQS , some don't

the debate is on whose opinion occurs in practice

like 20 people want EOQS

A plate of momos

11:05 PM

concentrated mobilized troops

tactical gureilla warfare

deleting stuff in 2013 was unthinkable

unless it was spam

A plate of momos

how did it feel growing old with SE?

it has been like 9 years since then

I stopped using it at some point

A plate of momos

dayum

When I came back it was kind of different

A plate of momos

11:06 PM

country's gone

should've stayed to defend it :c

when I started using it

all of the greats where using it

A plate of momos

oh yeah

and posting left and right

giving no f's

A plate of momos

I think even physics stackexchange was great, there was even fields medal winner like Hooft and all there

nowadays not even 10% of the caliber answers are put as then

I don't really care about the caliber

I care more about the volume

A plate of momos

11:08 PM

so what you want is an anarchy?

Tons of questions, of all levels were getting answered

well yeah

A plate of momos

dayum

I think there should be an MSE political compass made

I deleted one of my questions from 2013 a year ago

A plate of momos

Bruh

I tried reading it and it made no sense

A plate of momos

11:09 PM

why?

Oh yeah happens to me a lot

but there was still answers from mathematicians

trying to make it work

A plate of momos

I think I have like some sort of disorder which just kills my attention in whatever I am doing like 8 mins into it

so it went from people trying to bring the best out of a question to trying to choose the best questions

I think what you're describing, something close to that is reddit, however even that is slowly tending towards MSE type policies though not as harshly

what do you think of reddit?

but reddit isn't made for answering math

here there was tags

there was mathjax

A plate of momos

right yeah

and it was obviously not allowed to post stuff that wasn't to the point

A plate of momos

11:11 PM

right

just enough policy but not too much

you couldn't upload a foto of some stuff

A plate of momos

muh fotos

like a library

it was kind of like mathoverflow

A plate of momos

I think in terms of making library of undergrad question it's pretty done

now

mathoverflow now is closer to mse before than mse now I think

people on mathoverflow are still brilliant I think

A plate of momos

11:12 PM

Oh yeah

so they aren't concerned with dunking on everyone

instead of trying to do math

A plate of momos

I posted a question few days back of application of a theorem, in MO I got an upvote in like 30 min then one person told it is not so complicated and I posted on MSE, then MSE I got slammed dunked by 3 dvs "simple theorem" type comment and a close

I think I deleted it

sounds about right

I remember some time there was a user with a number theory hypothesis he had come up with

and here he got dunked on

so I reformulated it and made it a bit more elegant

A plate of momos

did you repost it?

or like give back and made them do it?

and put it on MO and it got like 10 upvotes and a lot of insightful comments by great mathematicians

A plate of momos

11:16 PM

dayum very based of you my dude

based & mathematics pilled

sure

A plate of momos

people dving and closing questions as dupes as ones from 2015 immediately be like : youtube.com/watch?v=PPCYG271HyU

I think one of my first questions on MO was like a trivial result from first semester graph theory.

A plate of momos

I just checked your profile

you seem to have no MO account o_o

And I got an answer by Noam Elkies who is like a great mathematician in harvard

my mo account is hidden

A plate of momos

11:18 PM

dayum

you maybe terrance tao for all I knwo

Pretty far from that

A plate of momos

I wonder what will be the next big thing

like you know what will be the next MSE

what do you mean

Oh, that's interesting

I don't know

A plate of momos

I think while technology of real world is developing fast, the technology of asking and answering questions on internet about math has not evolved much

do you have conjectures?

yeah I agree

maybe discord?

it's fast

A plate of momos

11:21 PM

Well, I have a few points that I think should be addressed if any site would be made

there is a lot of movement on discord

the math subreddit has a discord

and it has a lot of volume

there was some dude in my uni that had a tiktok with a ton of subscribers

but then he got cancelled

A plate of momos

I'm confused

it was like a physics channel

A plate of momos

how does titok relate to next question and answer place

it can't

but it's still like a new "science divulgation" place

:)

A plate of momos

11:23 PM

Points which I think a succesor to MSE would have:

1. Inclusive for people of all community even those who have been under represented for long
2. Some sort of community built textbook / standard resources where people can be lead to
3. A blogging feature. Many MSE users want to keep a blog like place on their profile (has been requested on MO)
4. A way to write equations easily , as of right now, I think math jax is like an artifical difficulty. If you come to do math at a place, then your difficulty should not be knowing a programming language

Google translate is good enough to give translation of text in other language already so I feel like the translation part is already here

A cousin form to MSE which has feature 2,3 to some level is AOPs forums. However, that got diverged into discussions of Olympiad type issues

I disagree

wait

AOPS wasnt originally an olympiad style thing?

A plate of momos

Well I mean, while it was made in the context of that, it could have evolved to anything really

wasnt it founded by valentin vornicu?

A plate of momos

idk about exact past

but like all I know is that it has it's base features and anyone can use those features

I think it was sort of olympiad aimed

from the start

yeah ur right

although the books are kinda just money grabbers

as well as most of the courses

A plate of momos

11:27 PM

le capitalism

do you disagree with any points I said on which a succesor of MSE should have or do you ahve others to add?

I agree with those points

but I'm not really all that interested with helping noobs who screenshot questions

I still enjoy doing it

but I'm more interested in a place where people can post their stuff in blazing speed and upvote and downvote

A plate of momos

I mean, the software would auto translate the photo into a text

There is enough AI or computer vision technology to do that already I think

so like just a soup of thought

Have you seen in meta this discussion about FGITW ?

A plate of momos

bubbling and oozing

"Fastest gun in the west"

A plate of momos

11:30 PM

lol wut

yeah

A plate of momos

is that about who answers the fastest

you have to go a bit deeper into meta

A plate of momos

lol

deep dive

I have to see this

wait send me link

It was about a point in time when there were people who wanted to guide their students to proper understanding of stuff

and then there was ppl who just answered fast af

I don't have links

but there's a ton of posts talking about that

A plate of momos

11:31 PM

lmFAO

ahahaha

what has become of MSE now lol

I was more of a fan of the fastest gun in the west types

A plate of momos

you know something has gone wrong when the discussion changed from how to make the best answer to how to eliminate questions instead

well the thing is some people didn't want bad questions to get any answers

the thing is that originally you were supposed to upvote a question or an answer depending only on the question or the answer

A plate of momos

Is EOQ's in some sense like relegion?

it is trying to put morals on people on what they should do and shouldn't do

opposing the natural anarchistic state of the SE system

constituting of only ups and downs

originally SE was anarchistic

but then too much "bad stuff" popped up

and some people didn't know how to deal with that

A plate of momos

11:35 PM

maybe that is how relegion came into the world as well

have you seen the famous mysticial post?

A plate of momos

which one?

most of the mse stuff is downstream

from the so stuff

of course SO is more complicated

A plate of momos

oh

since it's a lot of ppls just wanting to make bucks

not so much understanding

A plate of momos

11:36 PM

I think so SO actually a bire more nice now

this is the post:

A plate of momos

bit

I asked a total noob programming question long time back and I got a lot of help and one person made the whole thing for me

553

A: Why is Stack Overflow so negative of late?

Why is Stack Overflow so negative of late? I hate to pile on, but I couldn't resist summarizing the problem. And I apologize ahead of time for any potentially offensive language that I might use. Basically there are 4 camps of users on Stack Overflow: The "caretakers" who want to keep the site...

A plate of momos

HOLY

OMG

I have an answer for your category theory question now

he has drawn a wiring diagram from category theory to explain SE circumstance

xDDDD

inb4 I mistook what a wiring diagram is

sounds good

A plate of momos

11:39 PM

Certainly have

something similar is discussed in

David I Spivak's Introduction to category theory,page-78

tho I don't understand it enough now to explain it offhand

there was a video I saw which used a similar type of these linking diagram to understand racial inequality via category theory

invitation to category theory?

A plate of momos

Category Theory in Life - Eugenia Cheng

yes invitation

It's 2:42 am from me now. Should go take some rest. Was awesome talking to you though dude. It was really simulating to talk about state of MSE and your anarchistic standpoint on it xD

sleep tight

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