00:00 - 17:0017:00 - 23:00

12:08 AM
@Balarka cigarettes back on the menu due to my e-cig getting absolutely decimated

12:29 AM
I think the only thing of note personally for myself is that I figured out a square function set with its own sine, cosine, etc.
The function doesn't do angles, or at least I don't think it does without some temporal scalar function no doubt involving circular functions. It does, however, allow relating a unique location on a "unit square" of "radius" 1.

1:20 AM
Hi everyone
I have a quick question
Is anybody out here?

1:30 AM
Reading A Mathematician's Apology and Hardy seems like a bit of a prick

1:57 AM
@holahola That was a quick question.

2:34 AM
Does someone have a more constructive proof that a closed subgroup of a profinite group is again profinite? Something more enlightening than "A group is profinite iff Hausdorff, compact, totally disconnected, and a closed subgroup of a compact group is compact, so again profinite"

2:58 AM
Let $G = \varprojlim G_i$ be a profinite group, $H$ a closed subgroup of $G$, $\pi_i : G \to G_i$ the canonical projections. Suppose $x \in H$. Then $\pi_i x \in \pi_i H$ obviously, so $x \in \prod_i \pi_i H$ and the profiniteness of $G$ means the $\pi_i x$ satisfy the relevant inverse system so $x \in \varprojlim \pi_i H$..
Now let $x \in \varprojlim \pi_i H$ and let $x_i \in H$ be such that $\pi_i x = \pi_i x_i$.
err

3:39 AM
out here talking to myself because it's 5:40

4:05 AM
Yo @edward what's up
Also @ted!

Hiya

Doing the Amin strat of just shouting your math into the void

lol I think I'm getting there but I'm missing smth and i don't know what the smth is

Yeah maybe I'm just out of it but
I don't see it either
Thinking emoji

finite index crying
$\frac{\pi_i x}{\pi_i x_i} = 1$, cancel the $\pi_i$ so that $x = x_i$ qed
I listened to 10 hours of Skyrim - Music & Ambience - Day today
I have an idea
Let's say $y$ instead of $x_i$ because _ is too hard to type
Okay $\pi_i x = \pi_i y$ so $xy^{-1} \in \ker \pi_i$
does that even make sense
I showed that $H \subseteq \varprojlim \pi_i H$ so I can actually multiply those elements can't I ? Let's pretend I can
now $\pi_i : H \to \pi_i H$ is an open, surjective map, so $\varprojlim \pi_i H / \ker \pi_i \cong \pi_i H$, and $\pi_i H \leq \pi_i G$, and $\pi_i G$ is finite ($G$ is profinite so $\pi_i G$ is finite by definition)
err
right so $\pi_i H$ is finite, i.e. $\ker\pi_i$ has finite index in $\varprojlim \pi_i H$, so it is an open subgroup
I thought I had an idea but I've lost track of it
well $\ker\pi_i$ is an open neighbourhood of $1$, and $xy^{-1} \in \ker\pi_i$, and by the characterisation of the closure of $H$ that I recently found on google, since $\ker\pi_i \cap \overline{H}$ is non-empty, $xy^{-1} \in \overline{H} = H$ (since $H$ is closed) so $x \in Hy = H$ (since $y \in H$)
if I can't multiply $x$ and $y^{-1}$ then I can just say that $x \in y\ker\pi_i$ and $y\ker\pi_i \cap H$ is non-empty anyway so
Hence $H \supseteq \varprojlim \pi_i H$, so $H = \varprojlim \pi_i H$ is profinite
the end
idk if that works because I need that every open neighbourhood of $x$ meets $H$ for $x$ to be in $H$
$\pi_i y = \pi_i x$ means $y \in x\ker\pi_i$ and these are a basis of open neighbourhoods of $x$, so that gives me that $x \in H$
@Amin look upon my works and despair

5:14 AM
I have a very basic query regarding dimension of null and range spaces with regard to $Ax = y$ where $x$ is vector $(n\times 1)$ and $y$ is also vector $(m \times 1)$. So is it true that dimesion of range space is always $\le \text{min}(m, n)$ and also is it true dimension of null space = $\text{max}(m, n) - \text{dimension of range space}$ ??

Jesus lol

@jeea First, yes. Second, no. Dim of null space = $n-\text{rank}$.

@TedShifrin what is $n$ here? Supposing that $A$ will be $(m\times n)$
ok so null space is subset of $R^n$ because $x$ is of order $(n\times 1)$ ?

5:32 AM
Actually my Argument for the first Containment doesnt work; the projections arent the maps defining the inverse system

5:51 AM
@robjohn and all of the legs of right triangles are special cases in geometry

6:15 AM
ah I forgot to check that $\ker \pi_i$ is closed but that's true because the image of $\pi_i$ is Hausdorff
gonna shut up and go to bed now

cya, pal

6:45 AM
\o @Secret

hi

what's up with the new avatar? pal
Lain Iwakura?

7:04 AM
nah
just some random vTuber character thing

3 hours later…
10:00 AM
can someone help me with this
1

Let $\boldsymbol{X}$ be a random vector in $\mathbb{R}^{m}$ such that no elements of $\boldsymbol{X}$ is linear combination of others, and $\mathbf{Y} \in \mathbb{R}^{n}$ another random vector. Then $\exists$ an $n \times m$ matrix $M$ such that (Covariance) $\operatorname{Cov}(\boldsymbol{X}, \b... 10:30 AM So, I've thought myself into a knot: A binary operation on a set$S$is a function from$S\times S\to S$, making it a relation on$S\times S\times S$. I've recently read somewhere that relation composition is associative, but I know that there are binary operations that are not associative, such as division, cross product, and set difference. What precisely am I missing? Is what I read about relation composition being associative incorrect? 10:40 AM Composition of relations is an operation between relations, what does that have to do with associativity of a specific binary operation? You might be thinking about (a-b)-c vs a-(b-c) But if subtraction is a composition here then, you actually have a+(-b-c) vs (a-b)-c which are the same and for cross product axbxc you have a(xb(xc)) vs (a(xb))xc which are the same also. In particular, the domain of axbx and bx should agree with that of c So, take a binary operation$\cdot$. My brain is telling me that I should be able to write the expression$a\cdot(b\cdot c)=(a\cdot b)\cdot c$in terms of composition of relations. Right? See, this is where the "thinking myself into a knot" comes into play. I keep confusing myself whenever I think about this I'm trying to piece together what the composition of relations would look like in this case, but I'm not sure if it makes sense Both$a\cdot (b\cdot c)$and$(a\cdot b)\cdot c$would be relations on$S\times S\times S\times S\times S$I guess these would end up being two different relations on that set, but how would they necessarily differ? 11:02 AM Ok so... C composed by B to give BC and then composed by A to give A(BC) A composed by B to give (AB) and then composed by C to give (AB)C However, B's domain already has C's image, so there is no difference in ordering (since whatever goes into A include C as well as the image of C is a subset of B) hence the relation composition do not lead to nonassociativity in the final image See this for a more formal presentation of what I just said the images and domains have to agree and this agreement ensures the associativity of composition In short, whatever operated by A on B also operated on the outcome when B operated on C thus the ordering that produces the image is always C to B to A, even if your order of composition differs 11:51 AM I am currently working on a problem from my operator theory course and the problem reduces to calculating the following integral over the unit disk$\Bbb{D}$: $$\int_{\Bbb{D}} \frac{(1-|w|^2)}{\overline{w} (1-z \overline{w})^{2+ \alpha}} dA(w),$$ where$dA$is the normalized area measure on$\Bbb{D}$. The problem doesn't state what the integral should evaluate to, but apparently it has a nice form...but I don't see anything at the moment. By the way,$z \in \Bbb{D}$is fixed. I tried polar coordinates, but that didn't seem helpful. 12:15 PM checks to see if anything I wrote at 6am made sense 12:56 PM You risk being disappointed Based on my experience with early morning math 1 hour later… 2:04 PM @AlessandroCodenotti tbf my only experiences with early morning math were when I was still up but I've had some pretty good 4am results and well 8am for exams, but that was always fine no For green people (not aliens) interested in AI. 2:37 PM Depends on the subgroup @feynhat @Alessandro I don't think I'm disappointed; I think the first argument actually does work hahaha How did you read my deleted messages Big Brother? Room owners can see removed messages LOL @Edward hahaha alright so I spent the entire night trying to prove the assertion of the first example of a book for which I need to read 170 pages rip are there any math books with no mathematical symbols in them? 2:45 PM The original manuscript of Euclid's elements probably so he just wrote everything in words if you don't count Greek letters as mathematical symbols I think mathematical symbolism is a fairly recent development Good morning! Good morning Ad Maiorem Dei Gloriam Ad Majorem Sathanas Gloriam - Gorgoroth 2:53 PM Yeah, I've read the Arabs wrote out all their equations in words. Using abbreviations for those words was the first step towards symbols. notation is better because of the psychological phenomena of "chunking." actually I don't know Apparently Diophantus would have written: $$\kappa^{\tilde{\nu}} \bar{\alpha} \varsigma^\prime \bar{\eta} \bigwedge \delta^{\tilde{\nu}} \bar{\epsilon} \mu^0 \bar{\alpha} \iota \varsigma^\prime \bar{\alpha}$$ for but notation seems to be better$(x^3 + 8x) - (5x^2 + 1) = x$@geocalc33 You're the first person to know what that stands for and tell me :) 2:57 PM hahahahaha that's surprisingly compact notation So, my thoughts are that the fact that I can get square sine and cosine only strengthen my conviction that I should be able to find sine and cosine for a circle. Pretty fun stuff: desmos.com/calculator/djkxiyi0tm Ancient Indian mathematicians used to write math in the form of poetry. Major results were usually condensed into a shloka, which is a verse consisting of two bars of 16 syllables each. Has there been much research on other types of periodic wave forms? @feynhat nice 3:00 PM @EdwardEvans Well do you have 170 nights available? Diagrams are a form of symbolization, no? @Alessandro I have probably until some time in march, so probably about 170 nights but in general there are more than 1 words on a given page I see no issue then 3:01 PM One of the well-known examples of such work is Bhaskar's Lilavati. (Fun fact: ICM's Lilavati award is named after this). I'm just not used to the topological arguments but that'll get better over the course of the seminar Whilst making love a necklace broke. A row of pearls mislaid. One sixth fell to the floor. One fifth upon the bed. The young woman saved one third of them. One tenth were caught by her lover. If six pearls remained upon the string How many pearls were there altogether? earliest use of the slang term "pearl necklace" maybe a bit too crude for this chat rofl wtf ed. looool why Feynman used to make-up his own symbols for calculus 3:07 PM Nice! Also, not quite sure why, but in the case of this square sine and cosine, 2 on the opposite side is a limit, and if I add more than two decimal places of 9s, desmos freaks out. @EdwardEvans It is clearly not used in the sense of that slang term. hahaha obviously not, but I found myself too funny not to share I'm just broadening the vocabulary of the non-native English speakers here assuming they don't already know it Correction: questionably broadening the vocabulary of non-native English speakers @EdwardEvans yeah good thing that books tend to get easier the farther in you are /s 3:10 PM lmao @user2103480 sad it is an important step to go from vocabulary to symbols :P @EdwardEvans topology is the beautiful part Modulus would like to know why it isn't on wikipedia's "List of periodic functions" @Alessandro the beautiful part is the arithmetic obviously but I forgive your lapse in judgement 3:12 PM Pffff Built a nonassociative division by zero algebra. The bad news: Most of the undesirable features from the associative cases that is worked on back in 2015 are still there mostly, in particular you can only have x+x=x for any x The good news: Nonassociativity allows a lot of nice things to be retained, and can narrowly avoid having any null semigroups People who believe topology is the beautiful part are to be forgiven their transgressions, for they know not what they speak I am now reasonably convinced that the gist of these weird algebraic structures is their rock paper scissors property x+y=y and y+z=y never heard anyone say that arithmetic is the beautiful part of anything you literally just heard me say it 3:13 PM before okay now grrr Something something e^ipi beautiful something I thought the only reason for algebra was applying it to topology? Langlands said in his interview after receiving the Abel prize that he didn't really know why people describe mathematics as beautiful and then proceeded to avoid the question 3:15 PM nice I think the interviewers asked the question like 3 times and he just redirected them every time until they got the point sometimes I wonder too what exactly I find beautiful because its actually hard to come uo with examples which dont seem totally dry even for other mathematicians @user2103480 what is this beauty may be defined objectively as a measure of the magnitude of diversity of order in a given system harmonizing. or something like that. /shrug @LeakyNun a hot take 3:18 PM I think you realize something in Mathematics is beautiful not when you're in the process of doing it, but when you sit back and look where you started at and where you ended up with. Mathematics is beautiful when you're not doing it* Mathematics Imagine not doing mathematics @SayanChattopadhyay Nah, the process is everything, meaning the points when you are actually getting and synthesizing stuff Beauty is always in the eye of the beholder. 3:20 PM when it's finished it's just... meh ok this is another thing that worked now what about the 5492731 other things which are completely enigmatic What I mean to say that (personal opinion) every step that you do should seem like the most obvious conclusion, but those most obvious conclusions take you some place different. Mathematics is a beautiful little thing but pretty much everyone I know doesn't like it But I guess I find definitions to be the most interesting part of mathematics anyways I blame the schools for making everyone hate math and the lack of maths on TV that isn't for toddlers. I know by experience As Dyson puts it, some people are frogs, others are butterflies. 3:22 PM math is at its most beautiful when I wave my hands around like a maniac while trying to forcibly convince an innocent bystander that category theory is the answer to everything I look back and see that, in primary school, I used arithmetic synonymously with addition and subtraction, and that I now use it synonymously with classifying extensions of number fields Math is cool coz I get to draw cool symbols @Thorgott this is like that meme from it's always sunny in philadelphia where the guy is stood in front of a huge map with bloodshot eyes You know what's cool? I can draw a square!!!111 I like math 3:24 PM with a periodic function Math will be very beautiful soon for me, more or less as soon as they start paying me to think about it because I like to follow the evolution of it over time could I have a hint for the following problem? Show that if T is locally compact hausdorff, and H \subset T is locally compact, then H = A \ B for some A and B closed in T We should archive this discussion and put it forward as a list of justifications for calling mathematics beautiful lol ðŸ˜‚ 3:25 PM ~~hey does the chat support strikethrough?~~$\sout{no}$aw idk yes but using --- like this Oh oh lool There's a "help" link in the bottom right that you can consult for the formatting too 3:26 PM @Edward yeah lol relatable scenario @AlessandroCodenotti math will be beautiful as soon as I don't need to take dreadfully many courses to comprehend research topics anymore @user2103480 now you're speaking sense when you need to take a course to understand the title of a paper 4 Just become a programmer and never worry about formal documentation \o @robjohn I keep seeing that as robojohn 3:29 PM robo cop Robojohn is an auto-flushing toilet I'm going to listen to some periodic functions while I search for closed forms of periodic functions damn I'm just throwing out the aphorisms today @skullpatrol hey, whassup? Not much @robjohn how are you doing, sir? 3:32 PM so may I ask if there are methods of analysis that I might employ to reduce the error of my approximations to zero? I have one small idea, although it requires precision that desmos probably doesn't have. @skullpatrol still above ground There's a very tiny range of values for which I can cycle through that makes the error function zero at specific points at a non-linear rate. Any ideas? @AMDG an approximation to zero can be made with very little error. ;-) I don't get that one But I guess I see what you did there lol @EdwardEvans there's a papers such as "Generators of quantum stochastic flows and cyclic cohomology" and thats just so many buzzwords and I don't have a clue what that is some noncommutative geometry or so 3:36 PM I know "of" and "and" @AMDG If you would give us an idea of what you are doing, rather than asking for ways to compute functions with almost no error and not using series or recursion, then someone might be able to help. well, here, check this out: desmos.com/calculator/kfklfijqwg The green plot is an error function, and you can see that I've found an approximate maximum value of a_1 (see a_max) that between some minimum and maximum value of a_1, the value is zero. How is that computed? Desmos is probably using series or recursions or something similar to that to compute the exponentials there. Do you have some better way? Uh, not really, or at least I don't think so... also exponentials can be trivially computed (with integers) using simple bitshifts. given ln(x) = log_2(x)/log_2(e) ... You just provide an approximation of log_2(e) And then you can use exp to compute any exponential once you find a way to compute exp using one of its identities. So it would be silly, really, to use series or recursions Also I just learned you can use cosh(x) to compute any exponential just as well by using a constant of ln(N) as cosh(x*ln(N)) although I don't know the exact relationship between a^b and cosh(x * ln(N)) yet (would need to do a bit more digging). Wait, an approximation of$\log_2(e)$? How does that fit in with "no error" Each approximation will have to be made more accurate. That seems like using more terms in a series, and I though that was what you were getting away from. 3:51 PM log_2(e) is a constant. That form log_2(x)/log_2(e) is exactly the sort of form in principle that I'm looking for with regards to circular functions. If I want to increase the precision, I just provide a more accurate approximation of the constant log_2(e) :) I don't have to add more terms e is transcendental, so there'll never be an exact form apart from using the more efficient series definition of e as the limit of sum k=0,infinity 1/k! (It converges more quickly than (1 + 1/n)^n )$\log_2(x)$is not a constant, and even in binary, you need a series or recursion or something similar to compute it. Even with$\log_2(e)$, I thought you wanted something that you just needed to plug in a bigger$n$and it would give more precise approximation without series or recursion or the like. Even thought it is a constant,$\log_2(e)\$ to an arbitrary precision is a task that will probably require something that I believe you were trying to avoid.
I hope you find what you're looking for, but as far as I've seen, I don't see how it's possible.

I honestly don't get your point, and you can't assume that I'm using floating point math either. log_2(x) is trivially implemented as N - lzcnt(x) (count leading zeros) where N is the number of bits used to represent the integer x. I won't be using floating point as I'll be making my own rational number encoding that uses integer math only.
And I don't think you quite understand what I'm saying with regards to log_2(e) as a constant with arbitrary precision.

@AMDG I've written one of those, but computing logs in rational numbers is just as hard as with floating point.

the constant would be computed ahead of time with as much accuracy as desired.
Again, not using floating point math

@AMDG so it's not arbitrary precision.

4:03 PM
Rational number encoding that uses integers which doesn't suffer from the same error problems that FP math does?

@AMDG rational number approximations have built in error
they are approximations after all

Yes, well at least 3.0 is not like 3.0000000001 or something like that with ratios.

could I have a hint for the following problem? Show that if T is locally compact hausdorff, and H \subset T is locally compact, then H = A \ B for some A and B closed in T

@AMDG it's not in FP either. However, I know the limitations of a rational math package. I've written one before.

ive shown the other side of this implication (A\B is locally compact) but im drawing a blank with ideas on this side

4:06 PM
I'll be writing my own rational math package that will be very different than yours no doubt.
Also, this is well beyond the scope of reducing my approximation to zero.

@porridgemathematics compact implies closed, so locally compact implies locally closed

I just asked for any possible method of analysis
Not "you're approaching your problems that wrong way and I see an apparent contradiction in what you want versus what you are presenting"

Leaky Nun ah I see, thanks I'll try and play around with that

@AMDG Well, you seem to know enough about my rational math package, and I don't seem to be getting where you're going, so I am going to bow out from this discussion.

I don't understand. I just want to know what methods of analysis that I don't necessarily know about might exist for me to be able to get my error down to zero for my cosine approximation displayed there in desmos.

4:11 PM
Have you considered asking on the main site, Sir?

@robjohn I don't know anything about your rational math package, but on the contrary, you can't assume that mine will be like yours. You don't know what my encoding will be, and therefore the behaviors and constraints of what it will be.

hey guys
really sorry to interrupt
but any one has a clue as to why the frac lines aren't rendering on my chrome browser?
this is a screenshot of katex.org
(a part of katex.org)

@sai-kartik what latex are you using?
what's the source?

@robjohn sorry i dont know. How do I find out?

you might try right clicking on the rendered output (in the browser) and seeing if it shows the source

4:16 PM
@sai-kartik hello

the view page source option?

@skullpatrol I suppose I could do that, but is it really worthwhile enough to a be a whole question on math SE?

@sai-kartik how did you generate the output?

@robjohn I just opened the website
i didnt do anything else

@sai-kartik which website?

4:16 PM
katex.org

any particular page on that website?

@AMDG sure, it's worth a try :-)

@robjohn nope. just katex.org

4:18 PM
@sai-kartik get that damned book for one more day :-D

@sai-kartik Ha! it says "Server side rendering: KaTeX produces the same output regardless of browser or environment, so you can pre-render expressions using Node.js and send them as plain HTML." Obviously, that is not true.

@skullpatrol Excellent idea!
why didn't I think of that?

@robjohn not to be disrespectful, but that just went over my head. Can you please point out where exactly is the problem and if you have any suggestions to fix it?

@sai-kartik I was just quoting from katex.org about how browser independent their system is, but it's obviously not since you are having trouble.

4:21 PM
hmm
so it's just Katex's fault?
cus mathjax works excellently for me

@sai-kartik not sure

@robjohn oh..okay..
any way to ascertain the facts?

@sai-kartik Their page works for me, but I have no idea where the problem is

@robjohn no problem. Thanks for taking your time on this sir :)
but in the near/distant future, if an idea might strike as to what may have went wrong, please do ping me :)

@sai-kartik Okay, I am looking at their page.

4:25 PM
try here @sai-kartik

@skullpatrol sure thing! thanks :)

@sai-kartik Can you change the resolution/magnification on your browser? sometimes that affects rendering of thin lines.

@robjohn :o that seems to have done it !!

@sai-kartik At least I've been useful somewhere today! :-)

4:29 PM
@skullpatrol otherwise, so far today all I've been is a pain in the ass, it seems.

@sai-kartik Yay!

at 175%
@robjohn beautiful I must say!

that looks like a Chrome problem

yeah even microsoft edge..
firefox worked okay

4:30 PM
It should render those lines as 1 pixel at any resolution as far as I can see from looking at various resolutions in Firefox
@sai-kartik Firefox renders those thin lines as 1 pixel at any magnification

@robjohn you can't please all of the people all of the time, sir

@skullpatrol that's for sure
I just worry that AMDG is wasting time on his quest, but it's his quest, so I am trying not to interfere. I said my 200 cents worth and now I'm shutting the hell up.

@robjohn ahh
@skullpatrol true that

when your quest is for fire, you are bound to get burned

Firefox at a whopping 30%

4:36 PM
niiice

@sai-kartik Yeah, the lines are THICK because 1 pixel is HUGE at that magnification

im impressed honestly
@robjohn yeahh..
thank you, both @robjohn and @skullpatrol!
I'll catch you guys later!

@robjohn now I'm actually curious to know what your rational math package is like.
What I meant by "not FP" is that it isn't standard IEEE spec. My current idea to avoid ever growing spatial requirements for ratios is to use custom FP for numerators and denominators.
You'd just have some bits to specify where in the word the decimal place is. I could otherwise just use fixed point as well to simplify things even more.
At that point, it is the ratio of two ratios

@AMDG It started out as an arbitrary size integer math package (bigNum), then I added a rational type (bigFrac) and then built polynomials on top of that and specialized code for fixed point math using a common denominator. I also added strings for displaying things.

What kind of strings?

4:49 PM
I was going to incorporate FFT multiplication for large integers, but never got around to it

text or bit strings?

@AMDG Strings for displaying the numbers in different bases.
Text strings. The bit strings were already in the arbitrary sized integers

Oh ok

I haven't used that package in years, since the interface for Mathematica is easier to use and gives about the same functionality.

Wait a minute... so did you implement that thing in Python or something?

4:53 PM
@AMDG in C

sighs in relief

Interpretive languages would be kind of slow for that low level of a thing

Mathematica is nice... until my trial runs out of course in a few days :P

Like using a fork lift to hammer a nail in a board

Like using a spoon to stab your steak
The only use I have for mathematica is complex number plots because nothing else really does that for some reason except for giving some more or less useless "2D pretty plots".

4:59 PM
@AMDG I have lots of things I've written for Mathematica. The two most useful it turns out have been an Euler-Maclaurin Sum Formula implementation and a LaTeX rendering implementation.

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