This is the Realm of Simply Beautiful Art

I'm a bit of a late starter though. I was out of education after highschool for about a year and a half due to personal circumstances. Kept doing maths whenever I could. :)

And taught myself a little bit about programming and computers

user21820

Ah that's nice.

I always like the programming viewpoint of mathematical objects where possible, and it's a pity it's not very much shown in education.

shredalert

Yeah, I don't like how a lot of places split their teaching into pure and applied.

user21820

Like you know gcd right?

shredalert

12:20 PM

yeah

user21820

There are two equivalent definitions for integers. Some prefer "greatest common divisor", while some people use "minimum positive integer that is a linear combination".

The second one is non-constructive and non-intuitive.

The first is easy to construct via the Euclidean algorithm and prove that it really is the gcd and also that it satisfies the second characterization.

But I've not seen anyone else do it that way.

enderland

@heather I saw your post on Academia, I might be able to do some digging based on the university you listed if you want, just let me know

shredalert

@user21820 sounds like an interesting alternative way to do it

user21820

You know the Euclidean algorithm right?

shredalert

Yeah, can use it to show that any integer is of form ax+by=d

If I remember right

Simply Beautiful Art

12:28 PM

@enderland you'll have to wait until Heather is off of school

And you guys talk about weird things

@enderland Hello and welcome to my realm!

shredalert

@user21820 Is that what you meant by minimum positive integer that is a linear combination?

Simply Beautiful Art

We usually do math and random conversation, so you are welcome anytime

user21820

@shredalert Let g be the Euclidean algorithm. Then g terminates because the inputs' largest absolute value decreases on each step. Also g(0,x) = gcd(0,x) for any x, and we can prove that gcd(x,x+y) = gcd(x,y) for any x,y if x is nonzero, hence by induction g = gcd. Also, if the inputs to g are linear combinations of x,y, it makes only recursive calls on linear combinations, and the base case returns one of the inputs. So the output g(x,y) is always a linear combination of the inputs x,y.

It's clearly the minimum because gcd(x,y) necessarily divides any (integer) linear combination of x,y.

So done. And any student can execute the algorithm to see concretely how it works for himself/herself.

enderland

@SimplyBeautifulArt yeah I figured, I just wanted to post that before I forgot :P

@SimplyBeautifulArt I randomly picked the most recent chat room Heather posted in... which happened to be this one ;)

shredalert

@user21820 linear algebra and elementary number theory. But very concrete. Good way of doing it.

Simply Beautiful Art

12:34 PM

:-) @end

shredalert

@enderland but if you picked "the most recent chat room" how is that random? Hello, btw. :P

enderland

RFC 1149.5 specifies 4 as the standard IEEE-vetted random number.

3

Simply Beautiful Art

Lol

shredalert

Then "I randomly picked the most recent chat room" should have been "I randomly picked a chatroom" :P

Simply Beautiful Art

:P

user21820

12:36 PM

@shredalert: Or is it "I picked this room, chosen to be the most recent chat room, which is guaranteed to be random."?

Simply Beautiful Art

::rolls his eyes::

shredalert

I think it is "Of the most recent 4 chatrooms, I picked a random one."

if it is a 4sided die

enderland

I don't actually remember how I picked it tbh... now that I think about it

user21820

Hahaha!

shredalert

lol

Simply Beautiful Art

12:39 PM

::dies::

user21820

@enderland: You have spawned a random conversation about random picking.

shredalert

Funny how the chat description says "Room for totally random people to hang"

2

Simply Beautiful Art

How do you do something truly random?

user21820

Strange attractors here.

Simply Beautiful Art

@user21820 Lorenz attractor

user21820

12:40 PM

Yea it sounds better when I use the word "strange".

enderland

@SimplyBeautifulArt interestingly I had a middle schooler job shadow me a few weeks ago and that's one of the questions they were asking me, "how do random numbers work?" and... the answer is complicated :-)

shredalert

I think it has something to do with not being able to program something shorter than the description of the process

Simply Beautiful Art

XD cool

user21820

@enderland I was just going to say that. It's entirely possible that the universe is deterministic, in which case you're not going to be able to do something truly random.

shredalert

I vaguely remember reading about the definition of random in an expository paper by Kolmogorov

enderland

12:41 PM

it's even worse when you are programming since by definition it's reasonably deterministic

user21820

Well there are those RNGs that use sources like environment.

What we can say though is that some things sure look random.

@shredalert And one way to claim that is indeed via Kolmogorov complexity.

enderland

looking vs being are different though

user21820

Definitely, which is why I said it could be that everything is deterministic.

enderland

course, it doesn't matter if it is deterministic if it appears random enough :P

shredalert

I like Kolmogorov as a writer, he explains very lucidly

Simply Beautiful Art

12:46 PM

It sure does seem difficult to make a program produce "random" results

Oh wait, that reminds me. I think it would appear random if we used prime numbers

Say you have a four sided dice. Then take in some inputs, then take one prime per input (input 6 could be the sixth prime) and add these up, then mod4

That would probably look random

Hm, more even than odd...

Or something....

Perhaps you could ignore two as a prime and take mod 5 instead of mod 4

You guys know how to do this?

shredalert

1:04 PM

At long last, completeness!

user21820

@SimplyBeautifulArt: Actually primes have very weird non-random properties.

Like those prime races.

I think mod 3 and mod 4 give highly strange apparently non-random results.

The problem is that you can never prove something to be non-random.

@vrugtehagel: It's mathematically well-defined.

Hmm there are two main approaches, the first being axiomatic (say Kolmogorov's axioms plus extra stuff), and the second being directly founded on the foundational system (usually ZFC set theory).

In the first approach you simply stipulate rules for construction of random processes and deduction of their properties.

Which is pretty much how we intuitively reason about probability in common usages.

In the second approach you usually use measure theory to define probability spaces and then everything follows from that.

Of course, the first approach is only meaningful if we can find a structure that satisfies the axioms. That is where the second approach is needed anyway, even if when we actually do probability we stick to the first approach most of the time.

In both approaches, primes are completely determined and hence non-random.

But there is a way of capturing the notion we wanted earlier.

It's cryptographic randomness.

The ability to predict the next bit in a bit-stream given the preceding n bits.

A cryptographic PRNG is random in this sense if you need a crazy number of previous bits to be able to determine the next bit.

We call these cryptographically secure PRNGs.

More or less.

So if you take the sequence of primes and do something with it to produce a stream, the question becomes "How many bits do you need to observe to predict the next bit better than guessing?".

But there are also practical constraints; it's not likely that you have the computational resources to start at a crazy large prime, and the prime races show that at least in the first billion or so primes the modulos seem not quite random.

Cryptographically secure pseudorandom number generator

A cryptographically secure pseudo-random number generator (CSPRNG) or cryptographic pseudo-random number generator (CPRNG) is a pseudo-random number generator (PRNG) with properties that make it suitable for use in cryptography. Many aspects of cryptography require random numbers, for example: key generation nonces one-time pads salts in certain signature schemes, including ECDSA, RSASSA-PSS The "quality" of the randomness required for these applications varies. For example, creating a nonce in some protocols needs only uniqueness. On the other hand, generation of a master key requires a higher...

shredalert

1:28 PM

Hmm, I'm thinking whether I should stick to this analysis book or tackle one which uses metric spaces. I feel confident with the basic idea of sequences and limit now.

Yeah. I want to see what these fangled metric spaces are about.

Simply Beautiful Art

O.O

shredalert

1:43 PM

?

user21820

2:02 PM

@shredalert: Hmm what analysis book is that? Is it Spivak's?

Spivak's is excellent and I would recommend finishing it before going anywhere else.

shredalert

@user21820 The one I am using right now is called Numbers and Functions by R. P. Burn. I'm really not too fussed about studying loads of special functions and analytic things. I just want to have enough to get on with topology and differential forms.

Don't get me wrong, it is an excellent book, and really taught me well. But it's going in a very different direction than what I enjoy.

I found a very nice self-contained book by Robert Ash called real variables and basic metric space topology. It's exactly what I'm looking for.

It's available online. math.uiuc.edu/~r-ash/RV.html

user21820

I see.

shredalert

It starts off with some stuff on compactness and completeness, I was going to do completeness in my next chapter of Burn's book anyway. So it starts at just the right place.

user21820

Nice that you've already planned what you want to learn.

shredalert

Yeah, I'm a geometer through and through

I spent weeks looking for the right algebra book until I found the one that covers geometric algebra

and my group theory book is geometry oriented as well

That's the wonderful thing about the internet. We're so lucky to be able to look for all this information while studying.

user21820

2:42 PM

@shredalert Yes when I was in middle school I was using dial-up, so it was like super slow and not much academic resources available online.

You like Euclidean geometry problems?

shredalert

@user21820 I like the odd puzzle now and then :)

user21820

@shredalert: Here's one for the straightedge:

4

A: "Interesting" mathematical problems that are accessible to all disciplines?

Here is one. Give each one a piece of paper with a $2 \times 2$ unit square grid (namely $4$ unit squares) in the middle, and a ruler. Ask them to find a way to construct two points that are $\frac15$ units away from each other. There are infinitely many solutions, and you can pose the challenge ...

shredalert

@user21820 on first inspection I'm guessing instinct would say to start drawing diagonals

Brevan Ellefsen

2:58 PM

I've never been one of those people who enjoys studying fundamentals, such as ZFC. I might be be to list of the axioms, explain a few things about AOC and how it's tied to other things like Zorns's lemma, and maybe construct a few basic things, but it all seems so abstract to me. I seem to much prefer a high level, axiomatic approach to mathematics where I can neglect the underlying set theory

user21820

@shredalert: Try it! It's not hard, and even the solutions can be explained to the general public.

Euclidean geometry without the compass doesn't have much complexity.

Brevan Ellefsen

@user21820 hmmm.

user21820

@BrevanEllefsen: I also like an axiomatic approach to mathematics, which is why I've been advocating it somewhat over here.

Brevan Ellefsen

We lose roots, right?

user21820

@BrevanEllefsen: Are you familiar with the algebraic nature of compass-straightedge constructions?

Brevan Ellefsen

3:00 PM

Yes, a bit. I haven't attacked Galois Theory very hard yet

Or anything akin to it

user21820

Well so you can easily check that straightedge constructions can only solve linear systems.

Brevan Ellefsen

But I have the fundamentals

user21820

So if you start with points with rational coordinates you can only get more rational points.

The compass is what allows you to solve quadratics.

Brevan Ellefsen

Interesting. I actually haven't seen that before. I'll try to prove it later.

The straightedge only

user21820

Sure; it's a fun exercise to set up the definitions so that what I say can be formally proven.

shredalert

3:02 PM

@user21820 this reminds me of Courant's What is mathematics?

Brevan Ellefsen

Is there a hierarchy of what must be appended to Geometry to get another level of root?

user21820

It's not hard to do the same for the compass.. This means in terms of field theory that the set of constructible coordinates is in an algebraic extension of Q that is obtained by quadratic extensions.

This is how one can easily prove that 2^(1/3) is not constructible, by the tower law of field extensions.

@shredalert What reminds you? My post? =)

Brevan Ellefsen

I remember seeing that cubics can be solved with origami

user21820

Yes indeed, it's because they have a sliding operation.

shredalert

@user21820 yeah. I did a whole chapter on it

But that's over a year ago

user21820

3:04 PM

That operation can solve cubics. But by the same reasoning as before 2^(1/5) is not constructible.

@shredalert Ah I see. Well I try my best to find things that can be grasped by other people with the least prerequisites.

Brevan Ellefsen

@user21820 hmmm... I'll have to look back at my abstract algebra texts again soon and try to prove that. I set them down to study complex analysis more rigorously, as analysis is what I love most in mathematics thus far into my studies

shredalert

We can trisect the angle with origami

user21820

@shredalert That necessarily uses the same sliding operation, because cos(20deg) is not constructible.

It's the root of an irreducible cubic over the rationals.

shredalert

I know straightedge only gives rationale, but I don't remember the explanation

user21820

Basically the solution of any linear system of equations with rational coefficients is itself rational.

And that's all you can do with straight lines.

That doesn't help you find the minimum number of moves, however!

shredalert

3:08 PM

That book has fantastic soap bubble experiments

Brevan Ellefsen

Interesting. Given that this is something I've never thought of before, this might be a dumb question, but is there some tool that we can add to Euclidian Geometry to construct quintics? Is this also impossible?

shredalert

I'll give it a proper go tonight @user21820.

Brevan Ellefsen

I feel like it should be possible with a strong enough tool, but I've been fooled before

user21820

@BrevanEllefsen Not that I'm aware of. I never really bothered much because even the sliding operation is less 'constructive' than compass-straightedge.

So I just accepted that geometric constructions always miss some algebraic numbers. But I saw on Wikipedia that you only need to solve the Bring form of the quintic to be able to solve all others.

So if you have a geometric construction to do that you'd be done.

@shredalert Have fun!

shredalert

@user21820 I'm sure I will :p

Going to get started with that new analysis text after I get some lunch

Simply Beautiful Art

3:37 PM

@user21820 let's use the Veblen hierarchy and the fast growing hierarchy where the diagonalization step is changed to

$$f_\alpha(n) = f_{\alpha[n]+\beta}(n)$$

user21820

@SimplyBeautifulArt: For some fixed β? Then it vanishes for large α.

Simply Beautiful Art

Where $\beta = \max\{\varphi(a_1,a_2,a_3,\ldots,a_n):a_k\in\omega\cup\mathbb N_{<n},\alpha>\alpha[n]+\varphi(a_1,\ldots,a_n)\}$

user21820

Okay I'm going to have to read that. For a moment I thought you weren't going to say anything more.

Simply Beautiful Art

Sorry, on phone

user21820

No problem.

Simply Beautiful Art

3:44 PM

Sorry if my logic and stuff is bad

user21820

What is φ here? The multi-variable Veblen φ function?

Simply Beautiful Art

Yeah

user21820

I still don't quite understand your definition of β.

It parses as a valid set, but what is it supposed to mean?

Simply Beautiful Art

Oh, wait

I messed up

user21820

Ok.

Simply Beautiful Art

3:47 PM

It's supposed to keep the ordinal in the fast growing hierarchy from ending as soon as it does

user21820

It won't work like that because the FGH is defined transfinitely through all the ordinals.

Simply Beautiful Art

Expand on that for me?

user21820

At least, up to any computable one with a given notation.

Well you see the Wikipedia definition right? As long as you provide for each limit ordinal level some countable sequence, you can use that sequence.

For it to be computable, however, you need that sequence to be somehow computable.

So you need a nice enough notation for the ordinal itself.

Simply Beautiful Art

mhm

user21820

So that you can from the ordinal compute any given member of the sequence.

Furthermore, you want the sequence to be cofinal in the ordinal.

Meaning that the sup of the sequence is the ordinal itself.

This is what makes the limit level fast-growing.

Do you know what the ordinals and supremum are?

Simply Beautiful Art

3:51 PM

I think so

user21820

For example, to reach ε[0] you can use ω,ω^ω,ω^(ω^ω),...

Simply Beautiful Art

Yup

user21820

Now the FGH changes if you use a different sequence for limit ordinals.

Simply Beautiful Art

Yup

user21820

So you technically cannot just say FGH.

You must specify for each level the definition of the sequence.

That's what makes the whole business hard.

Because we need a uniform notation for all the levels you want to have in the FGH.

Simply Beautiful Art

3:53 PM

T_T when I say FGH, I usually mean, what was it, the Hardy hierarchy

user21820

Sorry I don't know the Hardy hierarchy.

Simply Beautiful Art

The one that goes up to epsilon not

user21820

That is Cantor's notation, and it's too weak for our purposes.

Using that indeed we can get up to f[ε[0]].

But we want to go further right?

If you have a particular ordinal notation for a computable ordinal k, then you can construct FGH up to just before k, and in most cases up to k itself as well.

Simply Beautiful Art

It is fine for my testing purposes

user21820

Uh I don't get what you mean then.

For example the notation I described to Deedlit goes way beyond that.

So the function I generated also.

Simply Beautiful Art

3:56 PM

I'm not using larger ordinals, I want to change FGH itself

user21820

I know. But the whole point of FGH is to reach all the way up.

Simply Beautiful Art

:-P

user21820

So you cannot stop at a small ordinal.

As I was saying:

> If you have a particular ordinal notation for a computable ordinal k, then you can construct FGH up to just before k, and in most cases up to k itself as well.

You should really think about why this is the case.

Your goal function is f[k]. Here k is a string representing the ordinal.

You use the notation to figure out the sequence for k.

So if the input is n, you compute k[n] (which you can for nice enough notation).

Then you recurse and compute f[k[n]](n).

It only works if the ordinal notation is nice all the way from 0 to k.

Simply Beautiful Art

But surely it is possible to have some g such that g[k](n) grows about as fast as f[m](n) for a much larger ordinal [m]

user21820

Eventually never, in a sense.

Simply Beautiful Art

4:03 PM

Is that because the difference between k and m become insignificant eventually?

user21820

Yes; I'm trying to say that eventually the notation that you use to describe k,m eventually will dominate the growth rate. But I can't precisely state it.

You're going to have to ask @Deedlit.

Simply Beautiful Art

Hm, okay then

One more thing

Imagine a set of functions. What if I recursively defined each function by higher functions? (Sorry if it is ill said)

With initial values of course

user21820

To try to make a more meaningful statement than my previous comments, suppose you fix a particular ordinal notation (which can reach up to <k), and you invent a new hierarchy g, then there are two cases. Either the notation you use to define g is less powerful than the notation for k, in which case the FGH will catch up with g before you reach k. Or the notation you use is more powerful, in which case you're better off using that notation to define an ordinal that you just plug into the FGH! =)

Simply Beautiful Art

Hm, I suppose I see

user21820

@SimplyBeautifulArt I'm not sure what you mean by higher functions. Computable stuff can only deal with computable functions on naturals/strings. So even ordinary functions have to be encodable as a program for it to be computable. Code is also a string, so you can indeed have computable higher-order functions.

Simply Beautiful Art

4:10 PM

I was thinking about an extension to Madore's ordinal collapsing function

user21820

Ah, what you would want to do those things is to simply find a syntactic way to encode those.

Simply Beautiful Art

::has large number nightmares coming::

user21820

You don't need to do anything else. Just find a way to syntactically encode the ordinals and find a reduction rule that can bring any ordinal in that notation to a lower ordinal such that you can reach any lower ordinal.

Simply Beautiful Art

No no, not really working with ordinals. Just plain functions

user21820

Then I don't really know what you're referring to.

Simply Beautiful Art

4:14 PM

:-) it's okay, I'll eventually understand what I'm saying as well. Give me a few years

user21820

Haha. Since I haven't looked really closely at ordinal collapsing functions just yet, maybe what you said will make more sense to Deedlit.

I like to have a real complete grasp of everything from ground up. So I go quite slow.

Simply Beautiful Art

Mhm, I imagine it will

"Slow" is not good for this context ;)

Well, I suppose it depends how you look at it.

user21820

My current understanding of how to generate fast-growing functions is more or less contained in my comment here. I now have a good grasp of everything up to the Large veblen ordinal. Either I will eventually reach the Bachmann-Howard ordinal or I will fail. In any case I should understand why I can or cannot reach it.

Simply Beautiful Art

Lol

You mean understanding it?

user21820

Well I don't know what to expect actually, because it's a natural stopping point and the last well-known ordinal before the ordinal for impredicative arithmetic. It's the ordinal for Kripke-Platek set theory, which is like ZFC but you can't construct sets that quantify over the universe. You can quantify over previously constructed sets.

Of course, if I were to follow the proof of its construction in ZFC, I should be able to understand it (syntactically). But to get a concrete understanding I can't just use ZFC.

There's a qualitative difference between being able to construct a large well-ordering in ZFC and being able to convince myself that a particular notation is well-ordered.

Simply Beautiful Art

4:25 PM

Cool

Yeah

user21820

I've been busy but soon I'll attempt to prove the well-ordering of the notation I described for the large Veblen ordinal, without using ordinals.

Simply Beautiful Art

@Phyllotactic so how many pages of paper has it taken to work out the FGH?

XD good luck man!

user21820

You'll see it when I get it! =)

Simply Beautiful Art

Lunch is about to end

=)

ヾ(´∀｀)ﾉ

user21820

Ok I'm going off to sleep, as usual.

See you next time!

shredalert

4:33 PM

goodnight @user21820

user21820

@shredalert: Good night! Tell me when you solve the geometry puzzle! =)

Simply Beautiful Art

@user21820 good night!

shredalert

@user21820 will do

@SimplyBeautifulArt enjoy the rest of the day at school

Simply Beautiful Art

:-P I'll try @shredalert

@user21820 hm, every ordinal notation is extendable through fixed points, isn't that how the veblen function is extended?

user21820

5:00 PM

@SimplyBeautifulArt Indeed, but you need a new notation to represent those fixed points. Also, it's easy to get fixed-points but it's slow. You need a completely different idea to go faster than before. Just to explain what I mean, consider that just getting common fixed-points of all previous functions gives the single-variable Veblen hierarchy. To beat that, one could take the common fixed-point of all those functions to get φ[Γ[0]], but then what?

Naively we could just repeat the process again, but that fails to reach φ[Γ[0]+Γ[0]] for the same reason as before.

So we instead take fixed-point of the notation itself!

Simply Beautiful Art

Ah, interesting

user21820

Of course to do so we had to come up with a completely new notation. So every time we want to go up faster we need a new idea.

I think the ordinal collapsing function sort of captures this idea of notation fixed-point.

And haha I'm really going to go off now! Just couldn't resist answering your question. =P

Simply Beautiful Art

=P thanks!

shredalert

7:56 PM

Proved the De Morgan laws for arbitrary sets, gosh it's been a while.

Simply Beautiful Art

8:29 PM

@everyone hello!

Hello!!!!!

@amWhy :O Hello!

Hello

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x: e i x = cos ⁡ x + i sin ⁡ x {\displaystyle e^{ix}=\cos x+i\sin x} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are...

Hm, this proof looks sketchy

@amWhy Close? math.stackexchange.com/questions/2195584/pi-0-it-cant-be

amWhy

Yeah...I think so....

Simply Beautiful Art

8:46 PM

I find the 6 answers to be hilarious. Sorry for saying so, but I'm still laughing. I mean, Dah! How many times does this need to be answered, and none the less, there are fish who bite the bait. — amWhy 1 min ago

Yup

The Count

to continue...

Simply Beautiful Art

@TheCount You are welcome to stop by here anytime by the way

The Count

long doesn't even begin to describe it. not bad, but just annoying. i am currently drying pages on the air conditioner.

thanks! i am not big on chat, but i do come in and poke around from time to time.

Simply Beautiful Art

Mmm...

If you are doing nothing, the transcript here is at least interesting, at least in my opinion

The Count

haha, that's true. it's why i pop around.

amWhy

8:49 PM

What happened? (that makes you have to dry pages on the air conditioner, and where are you that you have the air conditioning on?!

Simply Beautiful Art

@amWhy Spilled stuff on papers

@Lay Hello?

The Count

@amWhy I spilled soda on a stack of ungraded quizzes. :(

and the air conditioner is just on fan. i should have specified.

but still. too cold.

Simply Beautiful Art

Hm, @Lay does not have enough rep to chat

amWhy

@TheCount Well, you are creative in knowing how to dry off paper!

The Count

@amWhy it will dry them. it won't remove the color, the smell, or restore the shape of nice flat paper. so grading them will be a hoot. but my students think i am fairly silly in a good way, so i'm sure they will chuckle about it.

shredalert

8:53 PM

@TheCount putting paper towels over them and using an iron is good. On low heat

Simply Beautiful Art

@projectilemotion Howdy mate! (imagine a cowboy accent)

projectilemotion

@SimplyBeautifulArt How are you?

Simply Beautiful Art

Comfortably fine. Trying to see if I can come up with large ordinals right now

projectilemotion

@SimplyBeautifulArt Hmm, I see. I asked one of my friends to give me another one of these difficult differential equations.

Simply Beautiful Art

lol

The Count

8:57 PM

@shredalert i am hesitant because if i screw it up (and i am the kind of guy who spills soda on quizzes, mind you) then it would be a way bigger headache than just drying them and returning them like "oh jeeeeeez look at me".

but that is a good idea.

anyhow, i must be off. thesis won't write itself.

projectilemotion

Maybe I'll post it on MSE if I can't solve it.

shredalert

@TheCount If you are gentle it is very hard to mess up, but if you can see the writing on the next page air drying is good. Just make sure to put some dog ears so the pages don't stick

Saved more than one drenched novel :p

amWhy

@projectilemotion upvote!

projectilemotion

@amWhy Thank you very much!

amWhy

$\ddot \smile$

Simply Beautiful Art

9:10 PM

$\hat{\ddot{\ddot\smile}}$

amWhy

Is there anyone else here that needs to click on their "start chatjax" bookmark every time you use mathjax, or everytime you come to a chat, in order to see the mathjax render?

Simply Beautiful Art

everytime I enter chat/refresh the page

projectilemotion

@amWhy I do, I put it on my bookmarks though.

amWhy

Minor hassle, but I'd think I could train my browser to auto-launch chat-jax!

Simply Beautiful Art

xD

projectilemotion

9:14 PM

That would definitely help.

amWhy

I don't get it: two years after posting this answer of mine, I got a downvote today. No one has explained the downvotes I've "earned" there. Perhaps I didn't elaborate enough? But the OP seemed only to be looking for verification. Oh well!

Simply Beautiful Art

One downvote is clearly not focused on you

the other downvote is probably some angry person

projectilemotion

@amWhy It looks fine to me (+1).

amWhy

Oh well, I'd long forgotten it, until today's downvote. I've been receiving one or two downvotes a day, usually targeting older posts of mine. Not worth bringing to the mods attention. I guess it comes with the territory!

Simply Beautiful Art

:-/

$$C_0(\alpha,\nu,\mu)=\Omega_\nu \cup\lbrace0 \rbrace\\C_{n+1} (\alpha,\nu,\mu)= \lbrace\beta+ \gamma,\beta \gamma,\beta^\gamma,\psi( \delta,\varphi,\vartheta)| \beta, \gamma,\delta \in C_n(\alpha, \nu,\mu);\delta< \alpha, \nu\le\varphi,\mu \le\vartheta \rbrace\\ C(\alpha,\nu,\mu)=\bigcup_{n\in\mathbb N}C_n(\alpha, \nu,\mu)\\ \psi(\alpha, \nu,\mu)= \min\lbrace \beta|\beta \notin C(\alpha,\nu,\mu) \rbrace$$

Does this make any sense?

amWhy

9:30 PM

It's kind of sad, but I suppose there are folks here who feel the need to lash back sometimes. I remember about three years ago, in 2014, when there was an asker that had enough rep so that when 5 users closed a question of his, he'd in turn retaliate by downvoting an answer of every closer. (I think he ended up expelled), but at any rate, lost a lot of rep in doing so!

Simply Beautiful Art

$\Omega_\nu$ is some large uncountable ordinal.

projectilemotion

@amWhy This is one of my favorite answers of yours.

Simply Beautiful Art

xD

projectilemotion

Certainly doesn't imply that $x$ has slept with $z$ xD

amWhy

@projectilemotion That was probably the answer I had the most fun with, because I was simultaneously thinking of the relation in innocent and not so innocent terms! And in each translation, it worked perfectly!

Simply Beautiful Art

9:32 PM

Lmao

projectilemotion

@amWhy It's funny in many different ways.

What do you think about this one?

729

A: Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

$$\ \ \ \ \mathsf{W}\ \ \ \ $$

amWhy

9:44 PM

@projectilemotion Oh! That is a classic answer at MSE. I think I joined just before or just offer that answer was posted, and first learned of it when one of the regulars in $2011$ referred to it as "the W" answer....

projectilemotion

@amWhy Yes, I've heard it was quite a famous one after reading an off-topic meta post

amWhy

Kind of glad to see that this question is the most upvoted question on this site....

projectilemotion

@amWhy Why? Is that a joke?

amWhy

No, not a joke; I think the asker was very sincere and very idealistic. The fact that it was closed doesn't negate the interest it drew. It is ironic that when you look through the most highly ranked questions (and/or answers), it reveals a lot about the community, and sometimes reveals the limitations of purely analytic approaches to solutions.

projectilemotion

@amWhy Oh I see. It could be helpful for a lot of other people as well.

Simply Beautiful Art

9:59 PM

LMFAO That is hilarious! ::hats off::

amWhy

It also demonstrates how "soft questions" are either attacked, dismissed, closed, and deleted, OR, they somehow manage to strike a nerve that moves users here, sometimes generating a wide range of responses, and drawing in folks from other SE sites, etc. There's a lot of Sociological research about mathematicians and aspiring mathematicians that is fascinating.

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$Mathematics$ This is the Realm of Simply Beautiful…