serious-discussion

[storm sage]

yeah I can tell the spacing is pretty weird lol

[lost surge]

Another alternative is $\sum s^{(n-1)\dots 0}f^{(0\dots(n-1))}(0)$

[fathom swallow]

PhysMan

[storm sage]

it's nice though I agree

it can get confusing fast though

if you write the indices a lot

and in different places

(like subscripts, the denominator of a fraction, etc)

it's a nice way to visualize what's going on though

[blissful wigeon]

just to clarify to myself, differentiable is just a fancy way of saying the derivative graph is continuous, is it not?

[lost surge]

No.

A graph can be continuous but not differentiable.

[storm sage]

the derivative can be discontinuous

[lost surge]

Such as a cusp.

[blissful wigeon]

"derivative graph"

[lost surge]

Oh sorry.

[storm sage]

differentiable just means that the derivative exists

[blissful wigeon]

here's the derivative and value graphs of the two functions
https://www.desmos.com/calculator/frpkkohkkm

the blue one is the function itself

obviously

[storm sage]

if f is everywhere differentiable, f' is not necessarily continuous, but interestingly, it does satisfy the intermediate value property

[chilly hull]

oh its the opposite

nvm

[blissful wigeon]

is this not an example of a non-continuous yet differentiable graph

[chilly hull]

i read it wrong

[storm sage]

no, it's not differentiable at x=-1

desmos is probably just too dumb to display that though

lol

[blissful wigeon]

you said the derivative only had to exist

[storm sage]

yeah and the derivative of the blue function doesn't exist at x=-1

differentiable means that this limit exists:
$$f'(c) = \lim_{h\to 0} \frac{f(c+h) - f(c)}{h}$$

[fathom swallow]

Eric Tao (he/him)

[blissful wigeon]

just to clarify, they meet with a a ≥ and >

[storm sage]

yeah

[storm sage]

check for yourself that this limit does not exist when h → 0+

for c = -1

[neat lintel]

Are you going to give advanced ?

[blissful wigeon]

oh , that's what you mean

yeah, my mistake there, that makes more sense now

thanks

[wicked ore]

Can a function be differentiable but its derivative is not continuous? @storm sage

[arctic grove]

not this year no

[arctic grove]

[wicked ore]

Oop

Maybe that’s a stupid question

[neat lintel]

It’s a yes

[arctic grove]

I think the answer is yes

lets go

[neat lintel]

x^2sin(1/x)’s derivative at x=0 can be found with squeeze theorem

[wicked ore]

Jesus

Ok that’s not like

Obvious

Lol

[ancient flame]

LOL

[neat lintel]

It’s bounded above and below by x^2 and -x^2

[wicked ore]

lmao

[arctic grove]

it kinda is?

[wicked ore]

How

[arctic grove]

well its not something u can immediately think of probably

[wicked ore]

Yeah that’s what obvious means

[arctic grove]

but its what u can easily get to

[neat lintel]

It’s a limit technique

[arctic grove]

coz its got the perfect combo

[wicked ore]

I mean I was thinking an obvious counterexample would be a piecewise function but that doesn’t work

[arctic grove]

an oscillatory limit paired with a nice funciton

[wicked ore]

Yeah I have never seen that before

But that’s p cool

[neat lintel]

The idea is just f<=g<=h implies f’<=g’<=h’

So if you make x approach a value and the f’ and h’ are equal

You found g’

[wicked ore]

Ah

[wicked ore]

But why is this true

Like

-1 <= sin <= 1, but 0 <= cos <= 0 isn’t true

I feel like I’m making some obvious mistake though

[neat lintel]

Ah yeah

No you broke it apart pretty well

[wicked ore]

Hm

[neat lintel]

I need to restrict my hypothesis more

[arctic grove]

i think squeeze only works for limits

not derivatives

[neat lintel]

A derivative is a limit

[arctic grove]

coz i have never heard of a squeeze for derivatives

[neat lintel]

[arctic grove]

yeah i know

but still

[wicked ore]

He means it doesn’t work in that way for derivatives

[arctic grove]

yeah

[neat lintel]

I saw it pop up multiple times

I guess it’s just an introductory analysis thing

[wicked ore]

Ah

I haven’t done analysis yet

[neat lintel]

That’s actually what they start with

Feeding you counterexamples to unintuitive scenarios

[wicked ore]

Ah

[arctic grove]

[wicked ore]

Yeah I’d heard of the weierstrauss function that’s kinda insane

[arctic grove]

i have not

[wicked ore]

Seems like it’s doing a rigorous foundation for calc, but since you’re dealing with infinity it brings in a bunch of unintuitive baggage

[arctic grove]

its much more than that, at times u treat/build it all from the ground up

which can be painfully boring/too formalistic to be fun

but it has its good parts

[wicked ore]

Ah, even construction of the real numbers? How do you even do that lol

Ah

[arctic grove]

metric spaces

[neat lintel]

You can construct the real numbers in multiple ways

[wicked ore]

Cause rationals make sense, p/q set builder boom done

But real numbers…

[arctic grove]

dedekind cuts dont exist, dont @ me

[wicked ore]

Ah

[arctic grove]

R is the completion of Q as a metric space

[wicked ore]

Completion as a metric space?

[arctic grove]

that means every cauchy sequence converges to something in that space

yups

[neat lintel]

In fact for undergrad math EXPECT every intro to field X course to construct the real numbers in their own way. Its either algebraic, set theoretic, topological/analytic or something similar

[wicked ore]

Why would someone care about Cauchy sequences converging

What is a Cauchy sequence

[wicked ore]

Interesting

[arctic grove]

a sequence that goes smaller and smaller

thats not the rigorous definition ik

[wicked ore]

Oh a

[arctic grove]

but it gets the idea across well

[wicked ore]

I just looked it up

Pretty straightforward definition actually

[arctic grove]

yeah

very simple

[wicked ore]

Literally a sequence that should converge to a value

[arctic grove]

but they are extremely useful

and critical to get a grasp of

[wicked ore]

And it makes sense that you’d want your number line to have every sequence converge

[arctic grove]

no....

[wicked ore]

What why

[neat lintel]

It doesn’t uh…

[wicked ore]

Whoa

What

Lol

[neat lintel]

We call spaces in which every Cauchy sequence converges COMPLETE

the reals are complete

[arctic grove]

cauchy sequences are sequences in which the difference between consecutive terms can get arbitrarily small

[wicked ore]

Yeah I mean “should” in the sense of constructing a smooth, complete number line

[neat lintel]

Not consecutive…

[arctic grove]

oh yeah sorry

im jee pilled nvm that

[neat lintel]

No

[wicked ore]

No?

[neat lintel]

No

[wicked ore]

What about it isn’t

[deep mango]

[arctic grove]

[neat lintel]

Cauchy sequences say that any two elements in ur sequence can become arbitrarily close to one another for large enough N

That doesn’t mean those two values that become super close (to each other) converged to something

They could’ve diverged

[deep mango]

The differences between consecutive terms of the partial sums of the harmonic series converge to 0

But the partial sums of the series are not a cauchy sequence

[wicked ore]

Imma be honest I dont understand what this means

[arctic grove]

[neat lintel]

Just because element x got super close to element y as time tends to infinity doesn’t mean that x or y converged to something in the giant space it lives in. All we know is “x converged to y” in a sense. We don’t know if y converged in the set so we don’t know if x converged either

[arctic grove]

oh

[deep mango]

Cauchy sequence = for each epsilon, there's a point after which all elements of the sequence are within epsilon of each other

[wicked ore]

Yeah this is what I was thinking

[wicked ore]

I don’t understand, x and y are elements, they don’t get anywhere or converge to anything

Maybe I’m being dumb

[neat lintel]

Don’t get pedantic on me now

I’m being hand wavy for a reason

[wicked ore]

That’s not my intention

I just don’t get the explanation, that’s all

[neat lintel]

Let’s say you have two sequences x and y that are implied with indices

[wicked ore]

Ok

[neat lintel]

They get super close to each other as n grows super large right

How do you know the sequence x or y converged in the space tho?

All you know is the sequences converged in a sense to each other

But as a duo they might diverge relative to the space they live in

Blows up etc

Goes to an element not in the set

Think of a Cauchy sequence which converges to sqrt(2) as time goes to infinity

[arctic grove]

(or start oscillating between many elements)

[neat lintel]

But claim you only know about rationals

It’s definitely Cauchy

Did not converge in the set of rationals

So it’s a Cauchy sequence that diverged in Q

But it converged in R

[wicked ore]

Right that makes sense

[arctic grove]

[storm sage]

yup but the derivative always satisfies an intermediate value property

[arctic grove]

who knew geometers can explain analysis so well

[wicked ore]

But then

[arctic grove]

im complimenting u coocomba!

[storm sage]

so for example if f'(3) = 80 and f'(5) = 90 then somewhere between 3 < x < 5 we have f'(x) = 89

[neat lintel]

Topologists invented limit points!

[wicked ore]

What’s wrong with saying Cauchy sequences “should” converge, in the context of trying to construct a smooth number line?

[arctic grove]

So what!

this will not be considered topology by anyone

[wicked ore]

We know Q isn’t smooth because we have sequences that don’t hit anything, but they get arbitrarily close to a value

[arctic grove]

even if its topological

its used and first seen in the context of analysis

smh

[neat lintel]

That’s called completion. We usually take every Cauchy sequence and say STICK IT IN

if it wasn’t a limit point it is now!

The space is now complete

Spaces are not always complete

In fact we’d really like them to BE complete

[arctic grove]

they are very holey at times

[wicked ore]

Ah, so that’s literally

[arctic grove]

[wicked ore]

What complete means

Yeah

[arctic grove]

Homology moment

[neat lintel]

We have a formalism to make it happen

[arctic grove]

actually

[wicked ore]

Although I guess another more vague question is

[arctic grove]

is C complete as a metric space?

[neat lintel]

Yes it should be

[wicked ore]

How do we know the Cauchy sequence construction of real numbers works

As in, how do we know there’s no holes left, in a sense

[neat lintel]

This is why topology is important

[arctic grove]

thats where topo comes in

[neat lintel]

It turns out the completion of the reals is equivalent to its closure

[arctic grove]

the geometer can explain that better

[arctic grove]

Oooo

[neat lintel]

They are intimately related

[arctic grove]

thats sick

[neat lintel]

It is not true in general

[arctic grove]

i didnt know that

[arctic grove]

i see

[wicked ore]

Closure in topological terms

[neat lintel]

A closure says “find every last limit point of this set and put it in the set we have rn”

It’s equivalent to the set union its set of limit points

A completion focuses specifically on placing Cauchy sequences in the set

[wicked ore]

Ah

Wait I just looked up what a limit point is

[neat lintel]

Every complete metric space equals its closure im p sure

[wicked ore]

And it’s very similar

[neat lintel]

Let me check

[wicked ore]

It’s like if you generalized the “arbitrary closeness”

[neat lintel]

That’s a super strong claim

I got it

[wicked ore]

A limit point of S is a point in X such that every open neighborhood contains another element of S

[neat lintel]

Complete metric spaces must be closed but closed spaces need not be complete

[wicked ore]

Closed metric spaces need not be complete?

[storm sage]

what is a closed metric space

isn't every metric space by definition closed with respect to itself

[neat lintel]

A metric space is complete if and only if it is closed in every space containing it

[storm sage]

oh okay

that makes more logical sense

[full isle]

no such thing as a closed metric space

a closed subset of a metric space you mean

the proof roughly has to deal with looking at cauchy sequences

[velvet dagger]

Yeah this is one of the few things I don't like as much about Baby Rudin, it isn't clear on what properties are those of a metric space (so they hold for subsets by virtue of subspace topology) vs those which are properties of subsets of a metric space

Closedness is a property of a subset of a metric space, meaning if you want to say "A is closed", you must specify "as a subset of X".

While completeness is a property of a metric space

Now, if A is a subset of X, then you can think of A as a subspace, either in terms of open sets or in this case by restricting the metric

[bright hill]

Yeah but I don't believe in that definition

[mint canopy]

Don’t worry, that definition doesn’t believe in you either <3

[bright hill]

all you mofos sotruing me ain't got no shit

you tellin' me in the discrete topology every function is continuous?

get tf outta here, that shit ain't real

[vivid halo]

[arctic grove]

darq yes we are collectively lying to u!

we want to make u a bad mathematician

almost as bad as john gabriel!

[daring socket]

what txtbook

[tender tulip]

neat thing I noticed: For a subring $B$ of ring $A$, $B$ is a two sided ideal of $A$ if: $\forall a \in A, b \in A (a \in B \vee b \in B \Rightarrow ab \in B)$
But the conditional becomes a biconditional if $B$ is a prime ideal.

[fathom swallow]

Mizalign

[tender tulip]

actually if B is a subset of the A ring

[dapper badge]

But a subring B contains the same multiplicative identity as A, so if it were an ideal then it would be all of A

[tender tulip]

hm?

[dapper badge]

Nvm, I was responding to what you said before clarifying B is a subset, not a subring

[tender tulip]

now I’m even more confused

[dapper badge]

You said that if B is a subring of A, then B is a two sided ideal of A if this condition holds

But if B were a subring, it contains the multiplicative identity as A. The only ideal containing the identity is the whole ring, hence B would be equal to A

[tender tulip]

WHAT

[dapper badge]

Uhh, which part is "WHAT" worthy

Or is my brain on too little sleep rn

[tender tulip]

isn’t that for a field

also I thought an ideal is a subring

kinda like the ring theoretic analogue of a regular subgroup

[dapper badge]

Miz do you like for your rings to have multiplicative identities

[tender tulip]

yes

[dapper badge]

Then ideals are not subrings

If an ideal I of a ring A contains 1, then for all r in A, we have r1 is in I by definition of an ideal

Therefore r in I, hence I = R

Ideals are analogous to normal subgroups in the sense that the kernel of a homomorphism is an ideal

Ideals can also be thought of as submodules of the ring viewed as a module over itself

[dapper badge]

You might be confusing this with the fact that in a field, every non trivial ideal is the whole field

And that follows from the fact that every non zero element is invertible, hence xx^-1 = 1 would be in the ideal, which I've shown implies that the ideal is just the whole field

[maiden bear]

absolutely not

[dapper badge]

Timo I'm eating a yummy jalapeno and cheese bagel RIGHT NOW

[pure sun]

Did you bring enough to share with the whole class?

[dapper badge]

No I brought it for show and tell

No showing either though

[pure sun]

Haha

[maiden bear]

I was about to make a terrible topology joke with the bagel

Didn't let my intrusive thoughts win

[dapper badge]

why do seagulls fly over the sea

[maiden bear]

good question

makes me wonder what they would be called if they flew over the bay

[dapper badge]

probably S1 x D2 or something

[maiden bear]

this guys lost brother

[dapper badge]

he looks contractible tbh

[maiden bear]

he has to fit into the spaceships somehow after all

[dapper badge]

ig if it's a hyphen

R2 - D2

then its homotopy equivalent to circle

agony

timo you ever been rock climbing?

[maiden bear]

i have not

not the biggest fan of heights

[dapper badge]

I'm going for the first time today

kinda nervous, but I think it'll be fun

[maiden bear]

sounds nice walter

enjoy!

[pure sun]

Thats cool walter

I know lots of people who do it and they all love it

I hope you have a good time

[dapper badge]

my biggest fear is that it will rain and the rocks get wet and I slip and fall for several miles

which makes no sense because it's indoors

but I will update all of my fans on how it goes

[maiden bear]

They pour water from above for a more realistic experience

[pure sun]

Hahaha

I have a traumatic rock climbing memort from when i was a kid

I was at some kind of childrens museum that had a rock wall and i wanted to climb it

But like half way up my leg cramped and i fell off

[maiden bear]

agony

[pure sun]

And i was so embarrassed i started crying

And 20 years later i am still scared of rock climbing lol

[maiden bear]

Well i guess i did go climbing once

[dapper badge]

Thanks Buncho, I find great comfort in your story

[maiden bear]

which doesn't really count tho

[dapper badge]

jk jk lol, that really does suck though

[maiden bear]

i climbed up 2 thingies

looked down and went like

"hell no"

and quit

[dapper badge]

have you always had a fear of heights timo

[pure sun]

Hahaha i turned out (mostly) okay anyway :P

[maiden bear]

not really

it just randomly developed at some point

[dapper badge]

bizarre

[maiden bear]

i also despise elevators

always take the stairs

[dapper badge]

I guess it happens though, I only became scared of the ocean recently

but ofc there's nothing irrational about fear of heights or ocean

[maiden bear]

Yeah the ocean is scary as hell

[dapper badge]

there are those big squid people

or not people

squids

[maiden bear]

probably squid people as well

[surreal sapphire]

cephalopods are great

[dapper badge]

I suppose so

I go to the aquarium a good bit as like

exposure therapy

But now I think I'm just dependent on the glass barrier between me and the unknown

jellyfish look really pretty though, I'll give them that

[deep mango]

Rock climbing is so fun!

Are you going to do bouldering or top rope?

[maiden bear]

Aquariums are kinda cool but i always feel bad for the marine animals

because like

ocean very big

[dapper badge]

bouldering!

[maiden bear]

aquarium very small

[deep mango]

Bouldering is where you dont have a harness, it's shorter (like 15-20 feet) and there's big cushions on the ground

Ok nice

That's more fun imo

And also easier on the heights thing

[dapper badge]

It seemed like it, yeah

[maiden bear]

maybe i should give it another go some time

[deep mango]

I thought I was super into like

The idea of open ocean swimming

When I was swimming last week in the ocean at the beach

I was over a sandbar, and suddenly I came up to a 45 degree incline that just led into black

And something set off in my body and I just booked it in the opposite direction without even giving myself time to think

[dapper badge]

instinctual

[deep mango]

So I'm wondering now if I'd actually like open ocean swimming

Lol

But I still want to try!

[dapper badge]

yeah I've got massive respect for open ocean swimmers

[maiden bear]

I've been wanting to try stand up paddling

[deep mango]

That's fun

[maiden bear]

it looks like a relaxing activity

[dapper badge]

I used to do competitive when I was younger and thought I'd be able to do long distance open ocean when I was older

but it was simply not meant to be

[maiden bear]

there's this video of a stand up paddler getting his shit rocked by a dolphin

hilarious

[pure sun]

I kinda wanted to try rock climbing again but there is just nothing near me :(

[maiden bear]

Try that museum wall again

you'll get it this time

[pure sun]

The closest place (which might not even be open anymore) is like 40 minute transit each way

Hahahaha

I will crush the childrens museum rock wall

[maiden bear]

I believe in you

[pure sun]

I looked it up

Its 10-15 minutes away by car

But 40 minutes by public transit

The transit in my city is so bad

[hushed turret]

[pure sun]

I miss living in chicago

The bus and train system was great

Most people i knew didnt have a car

St louis is really car centric though

Which makes it hard

[dapper badge]

missouri, more like misery

living in/near an unwalkable city is very annoying though

[pure sun]

Yeah :S

I really dont want to learn to drive but i might have to at some point

I have anxiety about driving and anxiety about not driving and idk which one will win out hahaha

[dapper badge]

I think I get it, I'll go to any lengths not to drive in the city

I'm okay with suburbs, especially when the roads are empty

But I do still feel like driving is something most people would benefit from knowing

[pure sun]

Yeah i wont argue with that

[maiden bear]

I should get my license at some point as well

[pure sun]

I just dont want to like go through the hassle of learning and buying a car and paying for insurance and gas and maintenance

[dapper badge]

yeah owning a car is EXPENSIVE

[pure sun]

Yeah

The easiest solution would just be to date someone who can drive

But my bf also cant drive D:

So we are just double useless

[maiden bear]

My roommate drives and has a car

[pure sun]

Aww yeah

[maiden bear]

started to really appreciate that two days ago

because he ain't here atm

and i was shopping for a party that i hosted yesterday

carrying two of these was not that much fun

and usually we just got them into his car

[pure sun]

Oh yeah that sucks

[maiden bear]

i also live on the 4th floor

[pure sun]

Oh no hahaha

My grocery store is a 10 minute walk away and sometimes i buy too much and have a heavy load to carry

But i try to just go more frewuently and buy less at a time

[maiden bear]

Yeah i do the same

but sometimes i just get the urge to buy a lot of stuff and see how well i can carry it

[pure sun]

https://mobile.twitter.com/jamieloftushelp/status/1557097824927379457?s=21&t=MJDtp2WH8O8TdyG15yBhOg

jamie loftus 🏂 (@jamieloftusHELP)

ok i am rebranding my lifelong fear of driving and inability to get my license as an ‘ethical opposition to car culture’

Likes

47183

Retweets

5460

Twitter · Aug 9, 2022, 8:13 PM

Literally me

[maiden bear]

lmao

[wooden flax]

Lmfao

[storm sage]

BASED

[wooden flax]

Kinda hyped for my permit this fall ngl

[storm sage]

Okay but unironically car culture sucks

Yay for your permit though :)

[bright hill]

I'm an unironic r/fuckcars activist

[pure sun]

i followed r/fuckcars before it was cool

😎

[bright hill]

[tawdry falcon]

Yeah cars suck, but recommend a cool polyhedron to build with paper and explain why you think it's cool, plz

If I agree I might build it

[bright hill]

Cube

Coz it simple and cute

[tawdry falcon]

Yes, but I don't want simple and cute polyhedrons, I want cool polyhedrons

[tawdry falcon]

(I hope no one from the blockify cult reads this)

Also, I already built lots of cubes

[ocean thorn]

guys, i apreaceate all the helping you guys do, just try to explain as much as much in plain english while being specific about it. sometimes its hard to understand if one says something not entirely vague but a little vague. or one uses mathmatical notation or a=b something, while i understand that a=b, sometimes idk what to do with a=b if i haven't understood you well. thats all, apreciate it

[lunar spear]

You not gonna like me

appreciate

[ocean thorn]

think about it

[ocean thorn]

really think about it. really just. dream about this. because almost everyone who helps is uber pro max delta in math

Legend has it, @tawdry falcon is still typing to this day

[wicked ore]

Octahemioctahedron

No explanation needed

[tawdry falcon]

In most cases, people who help only give hints about how to solve a problem, or show how to solve a more general type of problems
You might not know what the person helping you takes for obvious and it's ok to not understand something instantly, it happens all the time when I ask for help. You should immediately ask them what they mean, to help them help you
Also, keep in mind that people who help do it out of love for maths and/or teaching (in fact, everyone does this for free), and no one will intentionally try to make things harder for no reason

[tawdry falcon]

That's a good one, seems tough to build with paper only.
Looks like it's made with triangles only, super cool!
Also, bonus points for cool name
:o all the triangles are the same size and form pyramids 🤯

[wicked ore]

Yep

I found it in an origami book so it’s certainly possible

[tawdry falcon]

Yes I'm working on it ^^

If I get it right I'll post it

[wicked ore]

Nice ok

[tawdry falcon]

I messed up some folds Some happy little accidents occurred, but I'm pretty sure hopefully I can get it right

[rustic island]

,calc 2 * 2^1023

[fathom swallow]

Result:

Infinity

[rustic island]

,calc 1.99999999999999 * 2^1023

[fathom swallow]

Result:

1.7976931348623e+308

[rustic island]

@hollow sundial for science ^

[bright hill]

,iam dying

[fathom swallow]

Gave you the studying! selfrole.

[chilly hull]

,calc 2^1024-1

[fathom swallow]

Result:

Infinity

[chilly hull]

,calc (2^1023)+((2^1023)-0.01)

[fathom swallow]

Result:

Infinity

[chilly hull]

,calc (2^1023)+((2^1022))

[fathom swallow]

Result:

1.3482698511467e+308

[tall badge]

#bots

[chilly hull]

oh yeah

[lunar spear]

nonna smart

and nice

[tawdry falcon]

Guys I did it ^^ It's beautiful (There's an ugly side, but I won't send it ahah)

There it is

(thanks @wicked ore for the suggestion)

[wicked ore]

Bro wth

Did you do that all by yourself

[tawdry falcon]

Yes! I copied a net I found online, if anyone wants it I can send it

[wicked ore]

Wow

That’s amazing

It looks great, well done dude

[wooden flax]

Lmfao

Looks amazing

[pure sun]

that is really cool. tbh i didnt think you were serious but that looks great!

thanks for sharing

[tawdry falcon]

thanks guys

[deep mango]

That is cool

[bright hill]

oh shit

this is sick

very well done

[tender tulip]

Czesaro polyhedron

[honest veldt]

Very pretty object, well done nonna

[neat lintel]

for this server, they should have a integration emoji

[sturdy surge]

I'm having a probability problem. Here it is.

There is a game that I am currently playing in which there exists different elements that can be controlled by a player. The "elements" are chosen by the 100 rolls, given to a player upon debut.

The chance of rolling spatial is 0.01% per roll. What are the chances of rolling spatial with the full 100 rolls?

Is it 1%?

[peak tide]

probability of NOT rolling spatial is 99.99% = 0.9999 per roll

probability of not rolling spatial in 100 rolls is 0.9999^100 = 0.990049338691372

[ancient flame]

#help-5

[peak tide]

therefore probability of rolling spatial at least once in 100 rolls is 1 - 0.990049338691372 = 0.00995066130862809 = 0.995066130862809%

so, very close to 1% but not quite exactly

this assumes of course that the rolls are statistically independent

[hollow sundial]

@austere hazel how goes school?

[austere hazel]

Frustratingly frustrating

Deserves to be said twice

That said i've had a bit of fun with the imc recently so there's that

[hollow sundial]

What stuff are you teaching/learning

Oh you were in Bulgaria?

[austere hazel]

Teaching, sets, logic, how to prove things, differentiation, integration, DEs

[austere hazel]

Nah, i just had fun with the questions

Trained a few who did go

[hollow sundial]

Oh nice

You competed vicariously

[austere hazel]

Is it really competing vicariously if i'm on the side of the ones making questions?

[hollow sundial]

Oh dang I didn't know you were a writer

Question maker?

[austere hazel]

I've always liked making questions better than solving them

[hollow sundial]

Takes a special evil to make questions

[austere hazel]

Part of the joy is seeing how many die and how many don't die

[hollow sundial]

Do you make questions deliberately using certain techniques like differentiating under the integral etc.

[austere hazel]

I like to combine techniques

[hollow sundial]

Impressive

[austere hazel]

Whether or not they turn out to be special techniques isn't all that important

[chilly hull]

i should make a diabolical calc test before i graduate hs as a funny joke

[austere hazel]

I support, i did that in high school

Got someone to take it even

[hollow sundial]

There's a sucker born every minute

[chilly hull]

30 questions

[austere hazel]

Later on i made several in both math and physics and had more suckers do them

[chilly hull]

actually no

uh

10 questions

[icy blaze]

Give it to your teacher and bet 1% of your grade they can't solve it. (Then maybe just slide a quick unsolved problem into it to guarantee your victory XD)

[chilly hull]

its going to be funny

[austere hazel]

Where's the fun in that?

[icy blaze]

Heck, I would have won that 1% if I had simply asked my high school math teacher what the square root of -1 was

[austere hazel]

...what?

[chilly hull]

i

[austere hazel]

I was taught that in middle school

[icy blaze]

I'm not kidding. I ABSOLUTLY COULD NOT convince my math teacher that imaginary numbers were a real usable concept in math

[austere hazel]

And right after that lesson i went to ask what's something raised to a complex power

[chilly hull]

how

idk about raising something to a complex power

but how do you not know sqrt(-1) as a math teacher

[icy blaze]

Complex powers generally just represent rotations in the complex plane, don't they? I haven't followed it too much, I'm more of a prime numbers guy than that stuff XD

[austere hazel]

Yes, but i didn't know that back then

[storm sage]

multiplying by e^i theta represents a rotation

in general, multiplying by a complex number means a stretch and a rotation

[icy blaze]

Yeah, that one I knew

[storm sage]

like for example if you multiply by 2, everything gets stretched by 2

[austere hazel]

Step 1: Look for teacher
Step 2: Hire random off street with fake cert
Step 3: ???
Step 4: Profit...?

[icy blaze]

The funniest part is

I literally showed her that the TI-84 calculators we used could work with complex numbers

And she still didn't believe that they meant anything at all XD

[storm sage]

RIP.

[peak tide]

step 1, don't require math teachers to have a math degree?

[austere hazel]

Is that how it works these days?

[hollow sundial]

Hahahahahhahahahaha

Hahahahahhahahahahahahaa

God no

[icy blaze]

Honestly, I'm not sure what worries me more- people who don't have certification to work the job they do, or people who DO have the certification and STILL have no idea what they're doing

[hollow sundial]

America K-12 education is pathetic

[chilly hull]

ok how do i create a document with latex

[austere hazel]

One of the reasons i'm thankful i'm not in America

[chilly hull]

like

looks like it was made in latecx

[icy blaze]

I'm Canadian, so it doesn't apply to me anyway XD

[dreamy siren]

People who teach sometimes get these sticks up their asses that they know everything there is to know in a subject and can't be questioned, its probably not that she doesnt know its i but rather that she refuses to believe concepts outside her expertise are more than little whimsical fantasies

[austere hazel]

You use a LaTeX editor

[peak tide]

"i graduated from high school, how can i not know everything i need to know to teach high school students?"

[icy blaze]

Fair point. She did kinda give me that "I'm the teacher, and what I say is true" vibe at times

I almost feel like it was a waste of my love of random math to write a whole paper on comparing the distributions of numbers with different amounts of prime factors with each other for the final in that class

[austere hazel]

Oh?

[icy blaze]

Because I know it went way over her head lol

[austere hazel]

I might be interested in having a look, if you don't mind

[icy blaze]

Ehhh, well it was actually very flawed

[austere hazel]

I have an ongoing side project i'm stuck in that seems to need a study of prime factor behaviour

[icy blaze]

The issue was, I based the analysis off of data from a program I made, but the program didn't properly count the number of factors a result had. BUT- I've done lots of other cool things with primes, if you want to chat about those lol

[austere hazel]

Why not, might be helpful

But erm, this seems like it needs more attention than i can afford today, so maybe tomorrow :/

[icy blaze]

I've actually been coding a prime factoring algorithm using my newest ideas. And alright

[austere hazel]

gonna add you as a friend so i can pm you~

[icy blaze]

Just ping and/or PM me whenever you want to know about it lol. It's nothing groundbreaking- at least not yet. But it's still a neat way to visualize them

ok

[austere hazel]

yeah, i'm not looking for groundbreaking, i'm looking for sufficient to get past that block i have

my main stuff is in FA and has little to do with primes

[icy blaze]

I see. Yeah, I just do math as a hobby, but I'm actually studying computer science lol. It's definitely nice whenever I need to make a program to do all the work for me XD

[austere hazel]

luckily (or unluckily) for me, everything i have to do is on paper xD

no computer coding necessary!

[icy blaze]

Yeah, there are definitely benefits to that kind of work. I mean, TECHNICALLY no coding is required for what I'm doing either

[austere hazel]

unless you count TeX as a coding language, which...i don't

[icy blaze]

I can just prove it much easier (Though, I suppose "prove" isn't the right word. Rather "apply the proof")

[austere hazel]

some days though, i feel like coding might have been a better option

[icy blaze]

Depends why you want it. I like to code video games, and want to make them for a living- it's just convenient to have for math as well

[austere hazel]

that reminds me i still have a visual novel scenario to write

i'm not worrying about the coding part until we get to said part

[icy blaze]

Makes sense. Python is a really easy language to learn, and has libraries available to make visual novels in (It's what Doki Doki Literature Club was made in, at least the original)

[chilly hull]

what else do i add

[icy blaze]

Not enough nested trig functions

[austere hazel]

please, why restrict yourself to sin and cos?

[storm sage]

chain rule is easy though

[deep mango]

[storm sage]

quotient rule is where it's at

[austere hazel]

why not tan, cot, sec, csc, sinh, cosh, etc

[chilly hull]

im making that question to waste time

[austere hazel]

you should put in harder questions

i'm not a fan of waste time questions

[deep mango]

Giv em a nested exponential or something

Make them do double log differentiation

[austere hazel]

integrate something that needs a non-obvious sub

[icy blaze]

besides, if you really want to waste time, just ask them to write the first 100 digits of some ridiculous fraction using long division. (I know that isn't calc, but like-)

[storm sage]

have them differentiate x x-arrow x

[austere hazel]

integrate something that needs induction

[chilly hull]

honestly i dont think i should make this test yet tbh

i dont think im at the point where i can devise diabolical questions on calc

[honest veldt]

devious calc 2 😈😈😈

[dreamy siren]

Get some 2d functions in there and throw in indicator functions, those will really fuck with people

[icy blaze]

Eh, just ask them what sqrt(-1) is. Worked for me XD

[chilly hull]

bro even i dont know what indicator functions are

ok yeah im not making it yet

not the time

[dreamy siren]

They indicate a point is a part of a set. 1 when yes 0 when no.

[swift sinew]

ryc, i-

[deep mango]

Hi alex

Did you enjoy the bb24 prejury

I did

I havent been paying a ton of attention this week

[swift sinew]

I’m two weeks behind

[austere hazel]

take your time to learn and ever devise more cruelly

[chilly hull]

i am going to make an immensely cruel calc 1 test on april

[hushed turret]

big brother?

[swift sinew]

I haven’t even seen ||pooch evicted||

[chilly hull]

[swift sinew]

I tried watching it on my tv but my housemates were being obstructive and I don’t have another way to watch it

[deep mango]

What the fuck

What???

[icy blaze]

Even I saw that. And it was the only clip of the only episode I've watched this season XD

[swift sinew]

I guess I could watch old episodes but that is so much time that I don’t have

[blazing pawn]

You dont have another way to watch it...

[deep mango]

You are

Light years behind

[blazing pawn]

it is literally 2022

[deep mango]

Ok just

Please please watch the week 3 veto and eviction episodes

You dont need to watch anything else

But those are 2 very very good episodes

[swift sinew]

Ok, I will do that - p sure I’ll get hooked from them

[deep mango]

Unless you know what's been happening since then

[swift sinew]

I don’t

[deep mango]

It's very fun bb

[swift sinew]

This better be worth it. I watched the entirety of last season in 2 days last summer, so I do have the stamina

[deep mango]

It's just 2 good episodes

[blazing pawn]

grad students: "yeah we're really busy, its nothing like undergrad"
grad students:

[swift sinew]

Well yeah but it’s never just 2 episodes

[deep mango]

Well sure

But like

[chilly hull]

how much harder is grad

compared to undergrad

[deep mango]

Way easier LOL

Uhh i mean

[swift sinew]

I’ll put it on tomorrow morning

[deep mango]

It sucks! We're all dying! Wah wah

[chilly hull]

on god?

[deep mango]

Yep

[icy blaze]

Shhh, don't reveal the secret! Fire everywhere!

[blazing pawn]

real

[deep mango]

Grad school gives you more free time but also you spend a lot more time doing harder math

And doing harder math takes more energy

[chilly hull]

so is like

[icy blaze]

I didn't have that question in my quizes. Must only be in some curriculums XD

[chilly hull]

you have more free time

[blazing pawn]

Not if you are fueled by rage and impotence

[chilly hull]

but the time you dont have free time is

[blazing pawn]

then it just leads to a feedback loop

[deep mango]

It's a lot about stamina and self control / managing your decisions well I would say

Lol

[burnt dune]

SA simpleart

[velvet dagger]

Main shtick about grad vs undergrad is local vs global pressure

Undergrad has a lot more structure

You have your weekly psets, you know your exam dates, at least have an idea how grading will look like

So doing that and just "living well" is kinda the correct course of action in undergrad. Maybe you have to think a little bit about what to do during summers for the sake of future grad school or career

Grad school starts off slightly like that but more chill

But then eventually it becomes a lot of global pressure but less local pressure/structure

Obv if you have a good advisor there's still a ton of guidance

But it's more like yeah make yourself look impressive enough to get good jobs by the time you're out, plus some mix of worrying about balancing prepping for possible academia and industry

There are some deadlines along the way but you have to have a lot more self control

There isn't a pset worth n% of your grade that's due this Friday, there's an advisor meeting which you should prepare for but which you can kinda wing if you're lucky and sometimes when you're not feeling it you take that option.... more times than you should

[deep mango]

This is a very good description of it

[burnt dune]

gamed enta ya daminark

[errant ridge]

idk why but without zooming in, your pfp looks like jesse pinkman

[icy crag]

#help-2

[ocean thorn]

Yea i do understand that. We all have our differences

[hollow sundial]

Starting to sound conplainy there. If you want your problem solved and aren't happy with the free help you're getting, go pay for a tutor

[ocean thorn]

No no, thats not the case at all. I didn't mean it in a negative manner. 😅 😁

I understand, appreciate, and respect all the amazing work you guys are doing. It was more about voicing the differences in levels of math we have in this server.

[ocean thorn]

And this statement was supposed to be a joke to emphasize the idea of the 'expert blind spot', the idea that since experts work on something for so long that many things are obvious to them compared to the layman.
The statement "dream about it" comes from how i feel sometimes, when i am between the phase of sleep and being awake, and in this dream like thinking; this is the phase where reality sort of breaks down in my mind. And everything starts to appear strange. Where normal and mundane things appear strange and don't make any sense. So assuming that it would be same with thinking about math, the parts of math that appear so normal often would appear strange in this phase.

Now back to doing math

[ocean thorn]

yea its communication

[dapper badge]

I am here to provide my update: I fell and died several times, but ultimately I survived. Overall I had a really great time and I'll absolutely go again when I have the chance. After we finished rock climbing we also hiked up a nearby mountain. Then we got greek food for dinner. A fun day for sure

also my new glasses came in

doctor's appointment in the morning went well and soon I will be on my way to the airport

[bright hill]

daily vlogs when?

also, please teach me how to die

lol

ya

I just started reading dandadan

that shit is soooo good

Seriously, @supple flame have you seen dandadan?

bruh

that's exactly what I'm thinking right now

worse yet, the manga is SICK

holy shit, are you even living

wuss dat?

[split island]

Hell’s paradise fan

I’ve just been there around the beginning of English serialization

[bright hill]

aww mannn

but the adaptation is gonna be from mappa!!

yohan, dis not fair

[neat lintel]

mappa is really just adapting everything huh

[bright hill]

hmm

it's not out yet

is that ok?

bet

[full isle]

adappa

[pure sun]

that's great!

[bright hill]

ah shit, I didn't take this into account

yohan, I'm gonna take my sweet fucking time with this one

don't you DARE spoil anything

bruh, my classmates spoiled so many shows for me

one I was talking to one of my classmates and I just heard about shigatsu no uso

so I was like, do you know this show

and I barely finished talking and he was like ||"you know the blonde dies by the end, right?"||

I'll never forgive them

I liked that show so much

[bright hill]

you can just read it from mangaplus, dumbo

they released all chapters for (all?) their mangas for free

even MY internet lets me read manga

jeez

[neat frost]

good morning rycie bycie!

[deep mango]

Ty for this, another show I am now relieved from ever watching

[neat frost]

Ryc have I complained to you about B trees yet?

[deep mango]

Not red black trees?

[bright hill]

you haven't watched your lie in april?

[neat frost]

No B trees

Stupid little shits

[deep mango]

What is B?

[neat frost]

Not specified

[deep mango]

I'm not a weeb

What is a B tree

[neat frost]

Like the creators never specified

[neat frost]

It’s a full tree (idk if that’s what you call it, but like all the leaves are on the same level) where you have an m, and every node has a sorted list of m-1 values with m children, and in each node except the root there must be at least m/2 values

You can then create algorithms to ensure that this remains full

Or complete

Whatever

So the height is always logarithmic

Anyway I just knew I’d fuck up a B tree question if it was on my test, and guess what? I fucking did

[bright hill]

some anime is mainstream

watching anime doesn't necessarily equal being a weeb

[neat frost]

It’s a generalization of a 2-3 tree

[deep mango]

Wtf

So dumb

[deep mango]

True

[neat frost]

Basically on my test I had to give an example 2-3 tree for a reason, and I forgot to add a fucking subtree. Like the root had 2 values and therefore should have 3 children, but I only drew 2

[deep mango]

Yes like spy x family and whatever else I've seen

But not whatever this lie in april thing is

[bright hill]

spy x family is barely mainstream

[neat frost]

I don’t think it would really have affected my answer, but that was like a large part of the question, and that was dumb of me

[bright hill]

BUT YOUR LIE IN APRIL IS, YOU UNCULTURED HEATHEN

[neat frost]

And the question was worth 12 points

[deep mango]

Slurp

[neat frost]

Yes

[deep mango]

If you dont get an A on the test

[neat frost]

RYC

[deep mango]

Call me and I'll beat them up

[neat frost]

ITS FUCKING B TREESSSSSS

I CANT

AHHHAHAHAHHHH

[neat frost]

Thank you rycie bycie

[deep mango]

Now let me tell darq whats what

[neat frost]

I fucking despise b trees

[deep mango]

THAT SHOW IS FOR. WEEEEBS.

[neat frost]

Now Imma wait for loch to return so I can complain to him

[deep mango]

Yes loch is more of an adult isnt he

[neat frost]

Also ryc I’m home and I have no clue where my family is

[neat frost]

No he just knows what b trees are lol

[deep mango]

Probably all getting ice cream without you

[neat frost]

So he can be like “oh don’t worry slurp that wasn’t a big deal, you’ll probably only lose like maximum 5 points”

And I’d believe him cuz he’s a CS dude

I just need the reassurance

[deep mango]

Oh dont worry slurp that wasnt a big deal, you'll probab

Fuck

I'm kind of a CS dude

[neat frost]

Ryc are you admitting that you’re a CS dude?

Omg he is

[deep mango]

I'm an everydude

[bright hill]

including an anime dude?

[deep mango]

You can be a math dude

I have no qualms with that

[fringe needle]

Anyone know how long you should wait for a mod mail request?

[neat frost]

I told you, you should’ve spammed ryc

[fringe needle]

I’m no spammer

[deep mango]

Oh yeah

One sec

I'm just checking in one more time and then we will get back to you

[fringe needle]

Thanks

[analog salmon]

Not a weeb btw

[fathom swallow]

gmod

[bright hill]

@ancient flame I don't think your definition works for finite sets

[ancient flame]

oh fuck

[bright hill]

besides, 0*inf is undefined

[ancient flame]

so then how would you show that $\int_{[a, b]}=\int_{(a, b)}$?

[fathom swallow]

gmod

[bright hill]

by using a stronger definition?

[ancient flame]

oh like lebesgue shit?

[neat lintel]

lowmath discovers measure theory

[bright hill]

lol

[ancient flame]

aw fuck

what like the singleton has measure 0 or some shit

yeah

idk

[bright hill]

oh

your definition is wrong in the first place

you don't multiply by 1/N, you multiply by (b-a)/N

[ancient flame]

fuck im dumb

I always make that mistake

[bright hill]

I should've noticed that sooner myself lol

[ancient flame]

icic

[frail lagoon]

Yo

[ancient flame]

alright thanks everyone

hi

[frail lagoon]

Well Riemann integration would typically be defined for real-valued functions anyway right so xd

But sure stuff like you can define some sequence going to infinity and keep everything else the same

[ancient flame]

lol tterra mutes chill

[frail lagoon]

Why would you not mute chill

[ancient flame]

bc I like chill

[frail lagoon]

The proverbial cesspit

[ancient flame]

LOL

[analog salmon]

That's my proverbial cesspool

[neat lintel]

did you expect anything else?

[bright hill]

also, this definition only works for sufficiently nice functions

I keep discovering things wrong with your definition

[ancient flame]

nope

[ancient flame]

but what was the point of posting this

[neat lintel]

look harder

i'll let you figure that one out

[ancient flame]

OH SHIT

LOL

2<1

[bright hill]

suppose you try to integrate the function $f(x)=x$ and the function
$$f(x)=
\begin{cases}
x & \text{x is rational} \
0 & \text{x is irrational}
\end{cases}
$$

[arctic grove]

Idk, mods seem to think its funny to rename their channels with higher numbers as if there are those many. shows how mature they are.

[fathom swallow]

DarQ

[ancient flame]

yeah that's not riemann integrable

[bright hill]

from, say, 0 to 1

[ancient flame]

alright yeah makes sense

I just had a 0 intelligence quotient moment

[sleek wing]

Meow

[bright hill]

yeah, by your definition, the integral of the second function is the same as the first

which is nonsense

[ancient flame]

woof

[open aspen]

That hurt my head to read

[bright hill]

[open aspen]

It was deleted

[bright hill]

oh

[hollow sundial]

oh yea i thought migi was talking about your $x$ x is rational tex

[fathom swallow]

riemann

[fading moon]

is there a machine learning channel here?

[deep mango]

no, but consider the related server linked in #old-network

[ancient flame]

gmod

[wild lantern]

Jee-mawd

[deep mango]

what

i've always pronounced it "guhMAHD"

but like

one syllable

[wild lantern]

Not like the game?

garry's mod <-> "Jee-Mawd"

[deep mango]

i didn't think they were related?

you know how there are gnu's

[ancient flame]

WTF

[ancient flame]

it's more like this

it doesn't come from the game tho

[deep mango]

what

[ancient flame]

it originally came from my irl name, and after being transformed and convoluted over the years it became gmod

[wild lantern]

Your real name is gregory mod

[ancient flame]

ofc

[bright hill]

I did no math today

I feel...

out of my depth

[vivid halo]

I should do some math today

[neat lintel]

it's okay to not do math

[vivid halo]

no

[bright hill]

no

[ancient flame]

bruh

[vivid halo]

nah like I have stuff I want to do just haven't had a good time focussing lately

[bright hill]

I went full monkey brain today bruh

I couldn't be productive for more than 2 minutes lol

nG, what math are you working on atm?

[ancient flame]

ignored

[bright hill]

[honest veldt]

it's fine Darq, just imagine a combination of words that look like they might possibly belong in the English language but you're not sure, and string them together

thats what nG would've answered with

[deep mango]

WTF

deepest lore

[vivid halo]

oop sorry I was relocating downstairs

[vivid halo]

splitting time between doing work on thesis problem and trying to finish editing papers

I've been confusing myself with hypergeometric functions lately

[bright hill]

what's it like, working on really hard problems?

open problems and the like

[vivid halo]

idk the problems I'm doing aren't actually that hard once you know what's going on, it's just writing everything up clearly and organizing everything is hard

or like, there's a lot of annoying struggling with getting notation and conventions from different sources to match up

[bronze pelican]

based

[bright hill]

yeah, that doesn't like something I wanna deal with

[vivid halo]

ehh I mean you have to deal with this to some degree with research

some fields are worse than others

also there's some tradeoff between like, is it easier to resolve conflicts like this or just work it out yourself

[bright hill]

work it out yourself?

like, make up your own notation?

[vivid halo]

that and like, reproving results rather than relying on or reproducing proofs that other people have written

[bright hill]

that could be fun, I think

@sleek wing why the deva? lol

[sleek wing]

dev

because that is literally me and sometimes not on purpose

[bright hill]

[tender tulip]

If we have a semigroup $S$ with a partial order R such that:
$(xy)Rx \vee (xy)Ry$. Let an element p be normal if for all x and y in S
$xRp \vee yRp \Leftrightarrow (xy)Rp$. let $Nor(S)$ be the set of prime elements. Then the anti-tail of an element $x$ is $K(x) = {y \in Nor(S) : x\cancel{R}y}$. Is the family of all anti-tails a topology?

[fathom swallow]

Mizalign
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tender tulip]

actually I wrote it down a bit differently

[bright hill]

mizlang, are you allergic to the advanced channels or smth?

[tender tulip]

banned from em lol

[bright hill]

why were you banned lmao

[tender tulip]

not using em correctly which is fair

Actually I'm gonna ask something different

Let $G$ be an infinite-order group, and $F$ be an uncountable collection of subgroups of $G$. Is the set theoretic intersection, aka: $\ \bigcupF = {x : \forall y \in F(x \in y)} \$ a subgroup? hhhh

[fathom swallow]

Mizalign
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[peak tide]

any intersection of subgroups is a subgroup

[tender tulip]

actually lol it is

if x is in all of them, then so must it's inverse be in all of them

[peak tide]

yep

[tender tulip]

and the identity is

now

[peak tide]

usual drill with intersections

[tender tulip]

what if we're considering ideals of a ring

[frozen merlin]

same thing

also same thing with closed subsets of a space

[peak tide]

any intersection of "collection of sets with property P" is usually a collection of sets with property P. Sigma algebras, topologies, etc

[tender tulip]

ah

The idea of a spectrum for a ring holds for any semigroup

[frozen merlin]

what's the analogue of prime ideals?

[tender tulip]

We just say for a semigroup $S$, an ideal is a subset $X$ of $S$ such that for all elements of $S$, if either $x$ or $y$ is in $X$, then it's product is in $X$. We say an ideal is normal or prime if this is biconditional.

[fathom swallow]

Mizalign

[tender tulip]

shit, gtg

[frozen merlin]

do you want your semigroup to be commutative?

[tender tulip]

Why does it matter

i’m referring to two sided ideals here btw

[frozen merlin]

the structure of ideals in rings is much nicer when they are commutative

[tender tulip]

ofc, it still generalizes tho

[frozen merlin]

true but I haven't heard of spec in non commutative algebra (maybe it does exist, not sure)

[tender tulip]

i think the set of prime ideals that do not contain a given ideal are elements of a topology of all of the aforementioned sets

it just happens to coincide

technically, the product of ideals is a product, so there’s a semigroup of purely ideals

[storm sage]

me when I read a proof: oh yeah this is super obvious
me five seconds later when I try to remember the reason: uhhhhhh

[swift sinew]

For me, not trying to remember the proof, but understanding the main ideas of the proof and why they make sense in that context are what help me “remember” proofs well after reading them.

[bright hill]

I find that trying to prove theorems before hand helps with this

just for like 10-15 minutes

I also do a writeup for particularly tricky/unintuitive proofs

[full isle]

what about it?

you can complete the semigroupnto a group ig

[dapper badge]

even ignoring desired topological properties, there are a couple of papers I've come across that illustrate why naive generalizations of Spec to noncommutative rings sorta just don't work