serious-discussion

[gusty sinew]

That doesn't have a finite subcover

[vast surge]

Yeah, exactly!

[gusty sinew]

You genuinely clarified like a billion questions I had

[vast surge]

Okay, so we've characterized open and compact sets

What are the closed sets?

[gusty sinew]

Empty set?

[vast surge]

Empty set is always closed, yes

Think of closed sets as complements of open sets

That is, if $G$ is open then $X\backslash G$ is closed

[fathom swallow]

Abelian Grapes

[gusty sinew]

Yeah

[vast surge]

So if I give you an arbitrary set F

My claim is that it's closed.

Do you see why?

[gusty sinew]

Yeah, because it's complement E\F will be open because it's a set and sets in the discrete metric are open so F itself is closed

[vast surge]

X\F, but yeah

[gusty sinew]

Sorry mb

[vast surge]

No worries

So, what we've shown is, with the discrete metric:
Every subset is open
Every subset is closed
Compact sets are exactly the finite sets

[gusty sinew]

Yep

[vast surge]

Any other properties worth exploring?

[gusty sinew]

Anything else you'd like me to think about?

Oh lol

[vast surge]

Do you know density?

[gusty sinew]

Uh let me think

Yes

[vast surge]

Okay. What are the dense subsets of X?

[gusty sinew]

Every point is a limit point => dense

None? Because for balls with radius smaller than 1, you'll have finitely many points in the ball

[vast surge]

Well there's one, but it's obnoxiously trivial

[gusty sinew]

Since there are no limit points you can't check density

Or does that mean vacuously true

[vast surge]

You're correct for all subsets of X but one

[gusty sinew]

Since you cant even check for density in the first place

Empt

[vast surge]

X is, of course, dense in itself

[gusty sinew]

Oh, thats a thing I've learnt too

Just forgot

[vast surge]

No worries

[gusty sinew]

I need to revise this stuff

[vast surge]

So our list of properties so far is:
Every subset is open
Every subset is closed
Compact sets are exactly the finite sets
The only dense subset of X is X

[gusty sinew]

I mean will i need this if I decided to do real analysis in the future? This topology seems to be just hyperformalising everything rn

[vast surge]

For the discrete metric in particular? No. It's ultimately not a very exciting place to do analysis.

However, taking a metric and asking these basic questions about the topology it generates is a useful skill

[gusty sinew]

Bump

[vast surge]

And there's a bunch of other properties we can ask about - T2, T3, T4, first countable, second countable, connected, etc.

[gusty sinew]

Not learnt that stuff yet

[vast surge]

It's also a matter of identifying which notions are useful in our specific case

[gusty sinew]

Mhm

Anyways, thanks for all the help

[vast surge]

No worries

I have an exercise you might be interested in

One sec

[gusty sinew]

Sure

[vast surge]

In addition to what Rudin is asking, for the ones that are metrics, I want you to try and describe the open, closed and compact sets.

And any other interesting properties you stumble upon along the way

Getting your hands dirty and working with a bunch of different metrics is the best way to build familiarity with this topic.

[gusty sinew]

Okay, will do. If I have questions, would a ping along with the question I have in #math-discussion be okay?

[vast surge]

👍

Anyway I need to buy groceries.

Bye

[gusty sinew]

Bye

[gusty sinew]

Alr so the first and second ones are just the Euclidean metric but squared or square rooted.so they should be metrics(you can sub in d1(x,y)=(d(x,y))^2 and d2(x,y)=sqrt(d(x,y)) to check triangle ineq. And ofc the other properties hold.).Coming to describing compact, closed and open sets, all sets with some property x (x=compactness or closeness or openness) in R with the Euclidean metric should also preserve that property x in these two new metric spaces, since we're just scaling our distances by some amount .The fourth one obviously isn't a metric due to the lack of symmetry and the distance b/w the same point being nonzero for all points except 0.

EDIT:The third one isn't a metric because d(-x,x)=0 as pointed out down below, and the fifth one is a metric, as we also figured out below. Il tell you about compactness and closedness and openness for the fifth one tomorrow, have to sleep rn.

Also the third one is actually like the Euclidean metric but you restrict your set from R to only R≥0,so triangle should hold

[tender tulip]

||Hint for 3: d_3(x,y) = |x - y| * |x + y|||

Fuck

[gusty sinew]

Yeah no i knew that

I just wanted to check triangle inequality

Im pretty sure it holds now

[tender tulip]

No,

[gusty sinew]

Oof

[tender tulip]

Wait nvm

[gusty sinew]

Whats wrong with this line of reasoning?

[tender tulip]

d(-x,x) = 0

[gusty sinew]

Oh ofc, annoying properties

[tender tulip]

yeppers

[gusty sinew]

Last one is 1-1/(1+d(x,y))

[tender tulip]

1/(1+1/d(x,y)) ye

[gusty sinew]

Time to check triangle

[tender tulip]

now sure how you'd check triangle ineq on that

nah wait

[gusty sinew]

I have an exotic idea, but idt itll help (binomially expand it)

[tender tulip]

i thought that immediately but idk if that would help

[gusty sinew]

1/(1+d(x,y)) = 1-d(x,y)+O(d(x,y)^2) so

1-(1/(1+d(x,y)))= d(x,y)-O(d(x,y)^2)hmm

The individual terms satisfy triangle, so the whole thing satisfies triangle, but I think now you want a proof of why (d(x,y))^n satisfies the triangle, but I think the easiest way to see it is by the binomial theorem: (d(x,y)+d(y,z))^n =d(x,y)^n+d(y,z)^n + (some nonnegative junk) ≤ d(x,y)^n+d(y,z)^n

[tender tulip]

by induction you could try to show triangle inequality on (d(x,y))^n and I suppose yeah

[gusty sinew]

Alr, about compactness etc for the fifth one, il be answering tomorrow

[leaden crown]

Is anyone good with math based on equations and/or graphing sine waves and sound based stuff? I'm taking a class in Acoustics and I'm struggling with the math.

[quasi jetty]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[gusty sinew]

A bit wishy washy i must admit, but my excuse is that I'm learning real analysis rn as a side thing whenever I get free time

[vast surge]

$d_{1}$ is not a metric. Consider $d_{1}(5,0)=25$, but $d_{1}(3,0)=9$ and $d_{1}(5,3)=4$.

[fathom swallow]

Abelian Grapes

[gusty sinew]

These are the consequences of wishy washy

[vast surge]

Saying "you can check the triangle inequality" is not a substitute for checking the triangle inequality.

Work it all out again, more rigorously.

[gusty sinew]

I just did in my mind, d(x,y)≤d(x,z)+d(z,y) and happily squared the inequality

d(x,y) is the Euclidean metric

[vast surge]

What's (d(x,z)+d(z,y))^2?

It's not d(x,z)^2+d(z,y)^2

[gusty sinew]

d(x,y)^2+d(y,z)^2+some nonnegative stuff

[vast surge]

What's (a+b)^2?

[gusty sinew]

A^2+b^2+2ab

[vast surge]

Right

[gusty sinew]

Which is smaller than or equal to d(x,y)^2+d(y,z)^2

[vast surge]

That doesn't matter.

If you're going to actually study real analysis, even just in your own time for fun, you need to use a pen and paper and work out everything that you claim, and make sure everything works. Inequality-wrangling can be finnicky and unexpected, especially if you're new to it, and so should not be handwaved.

[gusty sinew]

Yeah, currently trying to figure out where what I did fell apart

[vast surge]

There is an acceptable level of maturity where you can be handwavy with regards to these arguments; learning them for the first time is not that level.

[gusty sinew]

I did the triangle inequality in reverse: the correct statment is |x+y|≤|x|+|y|

[vast surge]

[gusty sinew]

Ive been trying to prove the following |x+y|≥|x|+|y|

Which is wrong

Il be sending you a readable document in a few days

Instead of a discord message

You don't have to check it, but il send it to make myself feel a bit better xD

[vast surge]

What book are you following for your self-study?

[gusty sinew]

I typed real analysis course filetype:pdf into Google and clicked on the first pdf, il have to check

Introduction to real analysis by William f. Trench

I should also say Im really doing this to understand what differentials are-for some reason when my teacher said x=u(t) => dx=u'(t)dt and I asked him what dx and dt meant on their own, he said they're just symbols so i went down the rabbit hole and apparently differential forms is the thing I'm supposed to get to

[prisma plinth]

Do you typically try to just look for a simple contradiction for these kinds of problems or do you just try to formally prove where it does fail

[vast surge]

Not familiar with it, give me a sec to look it up.

Ah, so you're near the end of the book already

[gusty sinew]

No haha what

I'd be quite happy if that were the case

[vast surge]

Metric spaces are chapter 8

Is this the right one? http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

[gusty sinew]

No, its like a scanned copy

[vast surge]

Okay, I didn't mean "is this the exact same copy", I was clearly asking if it was the correct book

Same table of contents, etc.?

[gusty sinew]

Yeah I know xD, still the book looks quite different

[gusty sinew]

Also, this stuff is at the end of the book

Differential forms

I know what something like dx/dt means, its nicely defined with a limit

[vast surge]

In that case you'll need to show me the book

[gusty sinew]

When I get to my laptop I will, also I can't send screenshots in any of these channels

[gusty sinew]

Forgot to credit youtube here too

I always go there when I don't understand something

[prisma plinth]

@vast surge hey homie for these kinds of analysis problems do you seek to immediately find a contradiction or do you try to formally prove it abides by some of the more tricky-to-show properties such as triangle inequality

since symmetry, strict positivity are kinda easy to show or contradict

[vast surge]

Depends.

For d_1 and d_2 I looked to find a contradiction, because they seem like at least one wouldn't work. d_5 looks like it should work.

[prisma plinth]

Like off the top of my head I have no clue how to show d2 or d5 abide or contradict triangle ineq

without just throwing pasta at the wall to see if it sticks

[gusty sinew]

Why does d_5 look like it works?

Its the most whacky one in my opinion

[prisma plinth]

i.e manually adding them and considering they are f(x) = sqrt(x) or 1/(1- 1/x) applied to d(x,y)

[gusty sinew]

The d(x,y) being in the deominator is just wierd

[storm sage]

Alternate between trying both

[prisma plinth]

I mean for the case of f(d(x,y)) you can try showing f(a) =< f(b) + f(c) if a =< b + c

[weak wren]

Does anyone know about Bayesian probabilities or bayes theorem

[jagged forge]

personally im going amd next time

the power draw & heat on the 14900k was more than i expected… had to upgrade to water cooling

[vivid halo]

https://imgur.com/GqowZiI

[plain owl]

I get very angry at content like that

[vivid halo]

lol

[plain owl]

people destroying expensive things

[vivid halo]

how do you know it's expensive

maybe it's bricked and worthless

and that's why they're fucking around with it lol

[plain owl]

potentially

but there is plenty of content that is purposefully flexing with new shit

[neat lintel]

real

but its so satisfying tho 😔

[ocean harbor]

google it

[dark obsidian]

Hello, i have just watched this video talking about the infinite hotel(https://youtu.be/OxGsU8oIWjY?si=qQB2ldXHqXDHbSQU) and i have a question about it. At the end he say that if you take a bus with an infinite amount of people with name that are a sequence of infinite A and B, you cant fit them all because there is a name which isnt in any room because he is different of all the names in every room of the hotel (sorry if i dont explain it well, i dont speak english very well maybe eatch the end of the video if u don’t understand)
But here is the problem i dont understand. If u replace A and B with 0 and 1, u can obtain some kind of binary numbers(0= AAAAAA…, 1=BAAAAA…, 2=ABAAAA… and so on. If u do that you can create a bijection between naturals numbers ( N ) and the names. So basically you could give everyone a room according to his name. I mean there is a bijection between the set of hotel rooms numbers and the set of people so it should be possible no? So the problem is that it contradict what the video say. Can someone explain me if im wrong or not?

[weak wren]

I did but in the help thing I need help for a calculation

[ripe needle]

Yeah its becoming an issue

[ocean harbor]

that's understandable but I don't know anything about this

so I can't help

[weak wren]

Oh ok

[weak wren]

Do you know someone who might know

[ocean harbor]

not really

[weak wren]

Ok

Ig I will just have to wait till that guy answers me now lol

[ocean harbor]

ye

[next schooner]

Diagonal argument

You cannot assign a binary number to each sequence like you say

[plain owl]

they don't correspond to natural numbers, you constructed an irrational

[plain owl]

you can, there are just more than naturals

[next schooner]

I mean binary integer

As they were setting out

[plain owl]

what's a binary integer

[next schooner]

A binary number that is an integer

[plain owl]

well the fundamental problem here is that binary number is not a thing, it's binary representation

so it's all about whether you have unique naturals or not

and whether you're even representing one

[next schooner]

Uh huh

[plain owl]

the argument looks like it's appealing to rationals while forgetting that those are in fact not rational

[next schooner]

I mean it looked to me like they were recasting each sequence as a binary number to obtain a corresponding natural number

No rationals involved

In any event, diagonal argument

Or construct 2-adics

For fun

[dark obsidian]

Thx for the explanation 👍🏻

[uncut turtle]

k

[ionic belfry]

Hello

[vestal silo]

did anyone do a project outside classes in first year, looking for project ideas that would actually help in some way

like idk, dont you need a portfolio or something for postgrad

classes are free rn so i have time

math major *

[sly bear]

😆

[surreal sapphire]

help in what?

[fervent flame]

some way

[surreal sapphire]

well, helping with academic career would just be learn more math

other than that, a programming portfolio (github) certainly wont hurt

[lean sinew]

quick question, if I go through calculus by Spivak, will i be able to go through real and complex analysis by Rudin? (i already took cal 1,2 but i need a refresher)

[tacit magnet]

No

Rudin RC starts with measure theory in a mmmm well I dont know how to express it

[tacit magnet]

Rudin is like this

If wanna study measure theory I recommend Bartle

Also, Idk if Spivak talks about Lebesgue integral

[woven whale]

i read the first two chapters of rudin and it was honestly what i was looking for

its in the style of a+m kinda

[plain owl]

what does that even mean

what do you mean by recasting a sequence

[vast surge]

No. Frankly even if you read baby Rudin first, I'd recommend reading something else before getting to daddy Rudin. Something like Royden-Fitzpatrick maybe, which spends a lot of time building up the theory of Lebesgue integration.

[alpine kindle]

why did you send a picture of serre lie theory

serre lie theory is so funny because there are 2 sections and each basically has the other as a prerequisite

[tacit magnet]

Lol

[woven whale]

average serre moment

[digital galleon]

Guys, quick question, can a pdf file contain a virus?

[woven whale]

yes

[digital galleon]

how?

[ocean harbor]

depends on the pdf

like don't download the pdf if it says a virus

[digital galleon]

Well, thanks for letting me in on the big secret lmao 🤣

[ocean harbor]

don't mention it lmao

[trail granite]

hi

[ocean harbor]

helo

helo is better than hello

https://cdn.discordapp.com/emojis/1048660781123780749.gif?v=1&size=48&quality=lossless

[oak stream]

Hey, why are these cycles not isomorphic?:
0 -> 1 -> 2 -> 3 -> 0
0 -> 2 -> 3 -> 1 -> 0

[chilly hull]

wdym "not isomorphic"

[oak stream]

do you know what graph isomorphism is?

[glass peak]

A graph isomorphism has to preserve connections between vertices

I’m guessing that your issue here is that 3 is not connected to 1 in the first cycle but is in the second, but the way you have written it is utterly unclear

[leaden surge]

Isomorphic means same structure, right? So those graphs wouldn’t be isomorphic since it has a different set of connections. Both graphs are a simple loop, but the nodes appear in a different order.

[oak stream]

Okay, but (2 -> 1 -> 0 -> 3 -> 2) is isomorphic to the original graph (0 -> 1 -> 2 -> 3 -> 0) right? Here, the nodes appear in a different order too. By the way, sorry for the late notice: the cycles are actually simple and not directed as I have written them. Sorry, I have been studying this whole night and I am wasted.

I will draw them in paint if it helps.

[glass peak]

Your connections are preserved in that case, it’s not about the order it’s the fact that 1 is connected to both 0 and 2

So in your first example where the graphs are not isomorphic you have an edge between 3 and 1 in the second cycle which doesn’t exist in the first

Hence there’s no isomorphism

[oak stream]

but isn't isomorphism all about renaming the vertices and preserving the degree sequence?

[glass peak]

An isomorphism is a bijection between vertices which preserves edges

[oak stream]

So is it not an automorphism but it could have been an isomorphism if we had another graph like 0 -> 2 -> 3 -> 1 -> 0?

automorphism: self-isomorphism

[glass peak]

I’m not sure I follow what you’re saying there, but a graph isomorphism is defined to be a bijection between the vertices such that f(u) and f(v) are adjacent iff u and v are adjacent

[oak stream]

Okay, so couldnt my bijection be like this then, (using 2 line notation):
0 2 3 1
0 1 2 3

[glass peak]

There is no connection between 2 and 0 in the second line but there is in the first, so no those graphs aren’t isomorphic

[oak stream]

Okay, got it.

thanks

[leaden surge]

Well, no, the nodes don’t actually appear in a different order. They are written in a different order, but the actual graph represented by your notation still has the nodes in the same order.

[naive bone]

uhmmmm HI?

[solar hawk]

omg ru guys doing graph ISOs

[sick karma]

I am going to be taking calculus III next year as a highschool student and am wondering what advice people can give me. Some questions I have: since calculus III is a semester class, what should I take as a follow up for the next semester; if anyone knows any unis or tech schools in Wisconsin that would be good for this sort of thing as an online class; and would it be better to take the class/classes at a uni or tech school.

[solar hawk]

do do isomorphic graphs have similar matrices

like the same matrix under difference bases similare

[glass peak]

If they offer a class in like intro. Diff geo or an analysis class that uses differential forms I’d recommend that, it follows in by showing you all of calc 3 in a much nicer theory

[sick karma]

Ok, I will consider that

[ripe needle]

If you haven't done Linear Algebra that that would be the follow up, it's normally taken just before, during, or right after Calc III...after that, Differential Equations is normally the 4th class in a calculus sequence. Which textbooks just depends on your personal level and knowledge, which you can check in #book-recommendations

It's always better to take at a university, but everywhere is different and it depends what you want to do. For example where I live, our community college offers Linear Algebra but it only transfer as a lower division class for applied students like engineers, the pure math major students at the university have their own upper division linear algebra course. Someone with better knowledge of Wisconsin might be able to answer that for you, you can also just stop in your local university and community college and ask advisors there what the best plan is. That's literally what they get paid to do.

[sick karma]

Thanks a ton 👍

[glass peak]

Without thinking about it too hard I would expect that isomorphic graphs would have similar adjacency matrices

[solar hawk]

yeah it sounds like something that would naturally arrise

maybe there is some graph to matrix theory field i have yet to see where they do problems like that

or maybe thats just graph theory

[glass peak]

You use matrices in graph theory quite a bit, you can describe graphs by their adjaceny and incidence matrices.

You also have matroid theory which is a bridge between graph theory (I believe more generally combinatorics but I’m not 100% sure) and LA. I only know the very basics about matroids because most of the section in my graph theory course was cut due to strikes

[solar hawk]

that sopunds very epic

ima search up matroid theory

[river moon]

indeed matroid theory is very cool

[glass peak]

Yeah it was cool, it was one of the only 2 sections in my graph theory course that felt like actual maths

[solar hawk]

lol

a graph is uniquely determined by its adj matrix right

whats the time complexity of checking similar matrices too what if u could use it to efficiently solve graph iso

[glass peak]

I imagine someone has thought of that lol

[solar hawk]

oh fair

LOL

me thinking i solved graph iso in P:

delulu af

[glass peak]

It is interesting to me that the time complexity of the graph iso problem is unknown though, It seems like everything else in graph theory is super easy or NP-complete

[solar hawk]

well i mean

there are bad complexity algorithms

so i guess it is known just not good 😭

[olive owl]

guys what comes first abstract alg or real analysis

[vivid halo]

they are pretty independent topics

either one before the other doesn't really matter

[slow jewel]

what is the difference between probability and statistics?

[slow jewel]

for some reason I notice a lot of people going calc --> real analysis when it comes to self studying

but not that much noise for studying abstract alg

[vivid halo]

I mean it makes sense, that's a natural progression

linear algebra -> abstract algebra is similarly natural

[visual breach]

People do not like linear algebra and idk why 😢

[bronze wedge]

linear algebra without proofs is... highly computational

(most people have different reasons i suspect)

[visual breach]

Imo 'Linear algebra without proof' is not linear algebra, it's just 'Multiplication table but for matrix'

[bronze wedge]

it still is "linear algebra", but by this logic "algebra" should only be reserved for "abstract algebra" etc

[visual breach]

Yeah I am not fond of pre-uni 'algebra' classes.

[leaden surge]

that's what computers are for

just please don't make me multiply a matrix by hand

just why

[bronze wedge]

i dont like doing computations, but it is not unreasonable to want your typical math undergrad to at least know how to invert a matrix by hand!

[leaden surge]

well yeah it's important to know algorithms I guess

although I say that from more of a computer science perspective

if I had to write a program right now to invert a matrix I could do it

[visual breach]

What I am sad with is that so many typical people despise proof-based linear algebra

[clear parcel]

I just called it multiplication with extra steps 😂

[visual breach]

hahaahaha

[clear parcel]

stuff about dual vector spaces and tensors is still linear algebra right? I did not know anything about those until recently.

[cinder zephyr]

yea

dual spaces come up everywhere

truely one of the most useful things to see

dual of a dual is the original space

[heady ermine]

Infinite-dimensional vector spaces be like

[cinder zephyr]

name one useful infinite dimensional vector space

[heady ermine]

L^2(anything)…

C(topological space)

[cinder zephyr]

ik ik there are a ton, was joking

[visual breach]

Wait, is C(X) notation coming from algtop

[heady ermine]

Amusingly, this is equal to its double (topological) dual

[visual breach]

Since continuous function X -> R can be regarded as a cycle C_0(X; R)

[heady ermine]

What.

Can it?

[visual breach]

Maybe not

[heady ermine]

What kind of cycle is this?

I essentially only know the kind in singular homology.

[deep mango]

Just c for continuous

[visual breach]

Hm

[deep mango]

Functions are cochains

Not chains

[visual breach]

Ah right, C^0 (X; R)

[deep mango]

Ya

[visual breach]

Something was strange ye

Still, same C >.>

[deep mango]

In that case, probably a coincidence

[fathom swallow]

Absta

[visual breach]

Caveat: Idk what the base ring is

[olive owl]

is set theory hard

[visual breach]

When is A-module equivalent to just a set, what A is this

[cinder zephyr]

if you've never done proof based math it can be

not because basic set theory is hard, but first learning proofs is hard

[visual breach]

@vivid halo, do you happen to know?

[vivid halo]

what

[visual breach]

Is there a ring A where Mod(A) is equivalent to category of sets?

[vivid halo]

uhh no

[visual breach]

Huh, why not

[vivid halo]

Set is not an Abelian category lol

[visual breach]

Oww, I was this close to describing C_0 (X; R) as a tensor product

[vivid halo]

lol

[winter cave]

https://youtu.be/ETkx7Ljyppk?si=MdxhmKN0kYZdtYeU

[proven karma]

Anyone here in 7th grade

I feel like I’m the only 7th grader here

[fringe summit]

Are you at least 13

[proven karma]

yea

im 14

i got moved down a grade tho

shouldve been in 8th by now

[split pawn]

Absta!

[split pawn]

It's okay

[proven karma]

thanks man

[split pawn]

You can move up a grade and even if u don't it's still okay

[vague peak]

Yo

Muhammed

[proven karma]

thank you

[vague peak]

ANYBODY TOOK THE AMC 8 MATH TEST YET?

[visual breach]

Oh, hi muhammad!

[split pawn]

Hi Absta

How you been

[visual breach]

Pretty busy recently

[split pawn]

SHOOT WE HAVE A SERVER RAID

[visual breach]

There are studies that I want to do

[split pawn]

See discussion

[visual breach]

So swift of an act

I barely saw it!

[sonic sentinel]

let's suppose, hypothetically, that me and my friend were chosen from the school for a selection exam for the country's international physics olympiad team. let's also clarify, in our hypothetical, that none of the grade 12 students, few of which actually studied an entire semester of physics, can attend because of age, and me and my friend have only studied one semester of physics (units 1-5 of ap physics 1). let's also suppose that this is happening ON the day of the last exam, so we're deferring one exam, and we have only one free day and like two hours at school to study. what would i do? what resources would i use to cram?

[fervent flame]

hypothetically, you would ask in a physics server

[sonic sentinel]

good call

oh mb didn't mean to ping

[honest badge]

Is there a rule for pinging people

[honest badge]

New episode of mission impossible

[plain owl]

why would there be?

[honest badge]

He asks for sorry

[surreal sapphire]

its just discord etiquette

[tribal slate]

theres no rule for being annoying too i think

not that i am implying anything i just only saw this sentence only

[quick beacon]

@neat lintel

Is there a limit to how long questions are allowed to stay up

Not to my knowledge

[surreal laurel]

just overheard some classmates saying the fundamental theorem of calculus is the most obvious thing ever, like “no sh—! Taking one step forward and one step back keeps you in the same place!”

we’re in week 1 of calc 2

[magic dawn]

Well that includes you proved that the integral is the inverse of the derivative (the proof of which is the fundamental theorem of calculus)

[solid snow]

Idk about “most obvious”

[surreal laurel]

i know

[solid snow]

It’s not apriori obvious that an antiderivative has anything to do with the limit of a sum

[surreal laurel]

school tends to treat integration and differentiation as inverse operators

so students look at the ftc like “duh! why do you need to spend a whole class on it!”

[magic dawn]

Tell your classmate to proof the ftc... that should make it obvious how obvious it is

[surreal laurel]

The flaw in my profs course outline is that he introduced the anti derivative before the idea of “taking the area under the curve”

[plain owl]

because they don't understand the actual definitions and only what they do

what sudo_lsroot said

[neat lintel]

Do I have to learn ftc to do these operations things quickly

I take a very long time especialy for quotient rule

[plain owl]

operation things?

being fast at calculating anti-derivatives is a practice thing and having a large bag of tricks

it's not really the same as good math theory knowledge

[scenic bridge]

I'm trying to determine what I'm doing wrong with my spending. I spent $24 today. Earlier I spent $10 on a cheesesteak and energy drinks. I must spend $8 a day on energy drinks.

Why shouldn't I factor in the energy drink on the equation for cheesesteak + energy drink in relation to "what is going wrong with my spending"?

[solid snow]

???

[plain owl]

that doesn't sound like a genuine question

[neat lintel]

So i dont have to understand the concept but I need to remember which number goes where using tricks that are not related to the actual theory?

[plain owl]

that's a bit vague for me to agree with that statement

understanding FTC is important

[solid snow]

knowing FTC doesn't impart a buff to your computation, but it's the bare minimum to grasp what it is that you're doing

[neat lintel]

Ah i cant send pics here

I sent it in #math-discussion

[scenic bridge]

Y = 2, B = 10, X = 24
B = 8 + Y
B = 10
X = 10 + B + 6
X = 24
Why can't I use Y and B to determine X?

[wicked zodiac]

Where are you pulling that random 10 from?

And 6?

[scenic bridge]

I spent $10

[wicked zodiac]

And I would assume thats B is it not?

[solid snow]

could you not just list what you bought and the price of each item

[scenic bridge]

It is

[solid snow]

no idea what you're trying to do

[wicked zodiac]

So why are you adding B twice?

[scenic bridge]

B is included in another equation

[solid snow]

total amount spent today = sum of cost of things you purchased

[wicked zodiac]

What do X, Y, and B represent?

[scenic bridge]

I'm trying to keep track of individual purchases and ascribe meaning, without looking at the big picture, so I can have an even bigger picture. Fuck.

[solid snow]

what does this even mean

its a sum

[scenic bridge]

There exists 3 sums
Sum 1, the sum of things purchased
Sum 2, the sum of an individual purchase

Sum 3, the sum of each individual object by category 1
Sum 4, the sum of indiviual object by category 2
Sum 5
Sum 6
Sum 7 ect

[solid snow]

???????????????????????????????????????????????????

[plain owl]

I can't believe this is an actual discussion

[solid snow]

adios brochacho

[plain owl]

don't get baited into this josemom2

[wicked zodiac]

Sum 1 = Sum 3 + Sum 4 + Sum 5 + ...

[scenic bridge]

The goat

[wicked zodiac]

[wicked zodiac]

I would assume "Sum" 2 would just fall under one of the categories

[neat lintel]

how do i find out what two numbers divid with a remainder of exactly 250?

Is that all the info u have

[visual breach]

let Sum n = Sum 1 + Sum 2 + Sum 4 in n
7

[ripe mango]

Help-28

[ocean harbor]

!noadvert

[quasi jetty]

Please do not advertise your help channel or thread in other parts of the server. There are many people who need help, so advertising can quickly turn into spam.

[versed axle]

how can i self study for math olympiad or other math tests

[sour timber]

help someone

Transformations of Functions (A1)

now

i have 57 minutes left

before my assignment closes

[quasi jetty]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[visual breach]

Also isn't this cheating

[neat lintel]

This guy is under the effects of drugs

[neat lintel]

Oh my god the new android discord is so trash

It's so bad just wow

The previous updates were just minor annoyances but now they've really messed things up

[scenic bridge]

They expand the mind apparently

[sage prawn]

Hello

[neat lintel]

I need an inexpensive / free "talking machine", a machine that either runs an LLM & TTS (best option), or uses an LLM API & runs TTS, or uses LLM & TTS API, + some programming, to "talk" as a charachter. The purpose of this is to provide such a device for my friend who is a sculptor. They will then create talking heads!

DM if you can provide programming or hardware or engineering (for battery, water proofing et cetera design).

[fervent flame]

have you considered gooogling

[glacial crystal]

Frlfrl

[neat lintel]

I can do that for $20/hr

I know to yap

[south arch]

hello, I am currently an undergrad student in uni and I'm considering switching majors to applied math. ik it's a really hard major though so I was wondering could anybody share their uni experiences so I can better understand what I'm in for?

[surreal sapphire]

do you have a syllabus you can share?

and whats your current major?

[south arch]

I'm a computer science student

I could probably share a couple core classes, give me a sec

off the top of my head, all the base math courses like diff eqs lin alg up to calc 3

ik there's a real analysis course, pdes, numerical methods, one other advanced course on diff eqs

a lot more I can't remember

[surreal sapphire]

mainly trying to get an idea about how proof-based the degree will be

because that would be the biggest adjustment probably

the classes you mentioned could go either way, except real analysis which i would expect to be proof based definitely

but i dont think its necessarily harder than CS, its just a different kind of thing

[south arch]

well to be more accurate I'm considering double majoring in cs and math, so I have a couple cs math classes I'm taking like discrete

[south arch]

really? I've heard otherwise lol

people complaining about the difficulty of math courses

[surreal sapphire]

sure, but those complaints mainly come from non-math majors in my experience

[south arch]

ah that's fair

[surreal sapphire]

i think a lot of it is just attitude

now getting used to proof-based mathematics takes a bit of time

maybe up to a year

so if you havent seen that and you want to get a taste, i would recommend reading a bit about it to get an idea

[south arch]

alright thanks, give me a couple mins I'll be able to send the math courses in the curriculum

[surreal sapphire]

(if you want to read an intro to proofs, i recommend https://math.hawaii.edu/~pavel/Aluffi_notes.pdf or alternatively i have written a small intro myself that is pinned in #proofs-and-logic)

[south arch]

ah, alright thanks

by proofs is it very different from the stuff you see in discrete math?

well actually reading thru a bit of that document, the intro is exactly what I learned in class so well see lol

[surreal sapphire]

ah ok nice

[fervent flame]

discrete usually is the intro to proofs for most students

[surreal sapphire]

this depends a lot on the class

discrete is sometimes used as an intro proofs

too slow...

so if you enjoyed that class, why not go for it

[south arch]

so here are the class names, lmk if u want further clarification bc some of these names are vague

diff eqs
linear algebra
applied numerical methods
introductory mathematical analysis
intro to pdes
intro to functions of a complex variable
methods of applied math
intermediate diff eqs
methods of applied math II
and a couple of electives I wanna take like maybe a grad course on applied statistics

these are the math courses I've yet to take

[fervent flame]

these are required for the program?

[south arch]

yeah

it's a double major in cs and math so I'm leaving out the cs courses

[fervent flame]

do you know if the linear algebra class is the same that pure math students would take? otherwise only the numerical methods, and 2 applied math classes seem potentially non proof based

well, diffeqs too

[south arch]

my college is an engineering college so def not

[surreal sapphire]

my main complaint with this would be its very analysis-heavy (not a single class in algebra for example)

do you have an idea of what kind of job you want to do later?

[fervent flame]

isn't this the norm for applied?

[south arch]

honestly, i would be fine with most jobs in CS, and most jobs that the applied math major would open like maybe something in fintech, another interest of mine was AI and ik R&D is hard to break into

but ive heard of undergrads doing it so

¯_(ツ)_/¯

btw, my college doesnt have a pure math major

[fervent flame]

both fintech and ml require a strong stats background

[south arch]

i might be able to take an elective, but is it very neccesary? and by algebra what classes do you mean? bc ive heard of scary things like abstract algebra, algebraic topology etc and i have no clue what those are and if they are good for applied math

[fervent flame]

ml specifically would be great to take optimization/modeling classes

[surreal sapphire]

i mean abstract algebra

it wont be super necessary for finance or ML

[south arch]

i was gonna cover the ML within my cs courses but yeah I should prolly take more stats courses

[surreal sapphire]

and if that is the goal, the applied math major will help

fwiw my background is degrees in math (pure though i guess) and computer science

but i work in roughly cryptography

[south arch]

ah, okay cool

what would you guys say i should expect from courseload, like how much time on hw

[surreal sapphire]

its weird to imagine a math major without at least a class in abstract algebra, but 🤷

[south arch]

bc ive heard of some demon hw problems which is one of the things that makes me a little nervous about switching

[surreal sapphire]

eh, probably ask other students at your college

[fervent flame]

thats both individual and school (and prof) dependent

[surreal sapphire]

i spent probably 10 hrs per week per class on homework

but i live on another continent so ...

[south arch]

oh wow

[fervent flame]

for me, most of my hw time was spent on hw for non math classes

i went to a no name liberal arts school fwiw

[south arch]

ah okay, so there is an applied abstract algebra class i could take as an elective

[surreal sapphire]

would probably recommend that for well roundedness

[south arch]

my school like many others im sure is known for weedout classes so thats the main thing that scares me

[surreal sapphire]

it might also help if you ever happen to care about cryptography or coding theory

[south arch]

i dont know a lot about cryptography or coding theory i dont think, but i am interested

thats why i said id prolly be fine with any job in the field, but id need to gain more experience and see

[fervent flame]

do you know which ones they are? for cs, id assume its DS/algo

[deep mango]

Coming from an analysis / applied math perspective i definitely recommend it as an elective, a lot of the ideas and methodologies are still useful

[fervent flame]

id imagine proof based LA or diffeq would be a math weeder, but im unaware

[deep mango]

Plus it only sounds scary, it's not actually that much crazier than the discrete math stuff

[south arch]

nah, the DSA course was fine, im taking one of the weedouts for cs rn so ill lyk (its a linux class and a computer architecture class)

[south arch]

ah alright, thanks for the advice

[south arch]

so just maybe like 2 or 3 main courses that are way more difficult in comparison with the rest of the curriculum

im familiar with those classes, its that im not familiar with any of the higher lvl math classes that makes me nervous

bc i dont wanna switch and realize im not cut out for this

[fervent flame]

did you do well in discrete?

[south arch]

im taking it rn, so far im enjoying it a lot

but then again its only been like 2 weeks

[fervent flame]

ah new semester right

[south arch]

yep

[fervent flame]

if you continue to enjoy it and do well, thats a good sign

[south arch]

all the math subjects and classes for the most part interest me, however im just worried thats not enough

alright, ill just see how the classes im taking go and do a bit more reading on the side about math to see if its what i wanna pursue

thanks for the advice everyone

[scenic bridge]

How can math be real if our eyes aren't real --Jaden Smith

[ocean harbor]

day 147 of hating java, let's go

https://cdn.discordapp.com/emojis/1048660781123780749.gif?v=1&size=48&quality=lossless

[neat lintel]

[solar hawk]

lol

[fervent flame]

i love overwriting pickle files when trying to read them

literally 2 days in a row

dont be lazy kids

[ocean harbor]

there won't be a java exam now yeeeeeeey!!

I cant imagine tell you how happy I am rn

[solar hawk]

lol

[ocean harbor]

https://tenor.com/view/swing-dance-swing-your-hips-dance-cute-dance-side-to-side-gif-25165719

[neat lintel]

in which year r u

[ocean harbor]

highschool

doing this credit for a uni

they call it ICS4U in canada

[neat lintel]

oooohh

thought u were in uni

[ocean harbor]

uni is gonna be tough i believe

haven't gotten any response from uni apps

except college

[neat lintel]

what r u planning to study in uni

[ocean harbor]

computer science and mathematics

[vivid halo]

tfw computer science jobs get completely replaced by GPT prompt engineering jobs by the time you graduate

[neat lintel]

Guys, if you learn coding/programming, then what will you be doing?

I mean what will you be coding?

[leaden surge]

I program things I want to

Mostly like physics simulations or other simulation related things

[ocean harbor]

building apps

or creating games

[leaden surge]

But can be anything, if you want to do it, and know how, and have the self-motivation

[neat lintel]

Oh okay

[neat lintel]

that’s it?

[ocean harbor]

like ive been interested in programming because I think it's a good career for me

[ocean harbor]

a lot more

[leaden surge]

“apps” and “games” are very broad categories

I’m planning to get a job as a programmer in a few years

[neat lintel]

I have absolutely no clue what to study

[leaden surge]

Dunno what I’ll work on

Basically just anything that’s not UI lol

[neat lintel]

My passion is gaming, but I’m really not keen on streaming or content creation

[ocean harbor]

good for you

@solar hawk

[leaden surge]

Idk if entertainment counts as a passion

I mean that would mean my passion is watching anime lol

But I guess if you’re really serious about games or something it can count

[neat lintel]

Wait how would you define passion?

[leaden surge]

Something you have an unusual or exceptional interest in such that you devote a lot of time to it, I think

Not a super accurate definition

I’m no good at describing words

Merriam-Webster has:

a strong liking or desire for or devotion to some activity, object, or concept

[neat lintel]

Oh then gaming is definitely my passion

[leaden surge]

I mean, how much do you game?

Is there anything exceptional you’ve done?

[neat lintel]

Currently i’ve quit gaming due to my studies, but when I do game I game for hours

[leaden surge]

Honestly I think that’s pretty normal

[neat lintel]

Hmm, well I have like over 2000 hours on a game called tf2

does that count?

[leaden surge]

A lot of people spend large portions of their free time on games

[leaden surge]

lol I have 800 hours there

Or maybe approaching 900

I forget

[neat lintel]

Oh

But I’ve been gaming tf2 ever since I was 8

made multiple accounts so i don’t know the exact number

[leaden surge]

I have like 10k+ hours on anime

[neat lintel]

That’s crazy

But yeah anyways, my passion is tf2 I guess

It’s the only game I’ve ever truly loved playing

can’t get enough of it

it’s like crack for me ya know

[leaden surge]

I’ve loved multiple games, but tf2 is the only one I’ve loved that’s designed for endless play

[neat lintel]

Yes, it’s a timeless masterpiece

[leaden surge]

Definitely a wonderful game

[ocean harbor]

weeb

[neat lintel]

I main spy, so I’m trying to reach a thousand hours on that class

Currently have 600 hours

[leaden surge]

Yes, akira

[neat lintel]

Still not on the level I want to be yet, but i’ll get there soon

[leaden surge]

(I can read hiragana/katakana lol)

[ocean harbor]

good for you

anyway thats what I want for programming

https://cdn.discordapp.com/emojis/1048660781123780749.gif?v=1&size=48&quality=lossless

[leaden surge]

What’s the three dots one?

[ocean harbor]

julia

[leaden surge]

And the atomic symbol?

[ocean harbor]

react

[leaden surge]

Seems like you like the languages good at data processing and number crunching

Whereas the market wants web-dev

Or generic c++

[ocean harbor]

ye

I've done cpp before but c no

aren't like same thing?

[leaden surge]

C is like C++ just fewer features

[ocean harbor]

ah I see

[leaden surge]

99% of C programs can successfully compile as C++, sometimes with minor modifications

[ocean harbor]

quite relatable

may I ask what languages do you know in programming

[leaden surge]

My favorite/most-used language is JS, simply because I find the browser to be the most convenient platform to develop for

[ocean harbor]

oh interesting

[leaden surge]

Mostly I make programs that display stuff on the html <canvas> element

I don’t really do anything UI related, usually

I also have some experience with C#, because of a school project I did that required Unity game engine

And right now I’m actually working on the server-side of a web app, and I chose python

Because the server uses apache, doesn’t have node.js installed, and no way am I using perl or php

[ocean harbor]

anyone can use python cuz python is always the best

languages ive done python, java, cpp, julia, react html, css and just a little bit js

[leaden surge]

I’ve also done a bit of Lua too

Because of a Minecraft mod called OpenComputers

Which uses Lua

[ocean harbor]

I did use lua in roblox studio but not that much

[leaden surge]

Also I have like a tiny amount of experience/knowledge of a handful of other miscellaneous languages, but not to any useful extent

Including C++, but it’s incredibly vast and I only know the very basics and the parts that are shared between most languages in general

Actually I did write a fluid simulation in cpp once

By modding an open source game that had an existing (but unrealistic) air simulation

[ocean harbor]

I did that too

but never made something good in cpp, only basic projects like integral area etc

i made a geometry dash game using java and took me a lot time to make it look good

I think it's my first time doing something that I don't like but it still looking good for me so i hope my teacher likes it too

#chill message

would you say anything about it? because I kinda feel the jumping isn't good enough lol

i've heard that in university they will teach python first year or c, I just need to get prepared for c tho

[leaden surge]

looks good, I guess

idk, I never played gemoetry dash

obviously level design could use some more... variety

might be a good idea to sync the level design with the music

[ocean harbor]

geometry dash game is already created by robtop so I decided to make this for my final proposal project then I realized that it's hard to make menu and stuff using JFrames in java but I guess this is my best I can do

at the first ive struggled making the sound work and it finally worked after a few days

pround of myself for doing it!!

[ocean harbor]

but yea thanks for this^^

[leaden surge]

I've struggled making my Linux laptop work lol

seriously, random things in linux just stubbornly don't work right

once you've got it working though it's generally pretty stable

[ocean harbor]

true

[leaden surge]

oh yeah unless you do something funny and break something

now my laptop can't fall asleep or else I have to reboot it

it's what I get for messing with nvidia drivers and shit

so I just disabled my computer's automatic fall asleep function

it just dims the screen and stops there

[solar hawk]

@light horizon im taking a shower RN

[fair mural]

[solar hawk]

Context is key

https://tenor.com/view/frog-gif-20986887

[fair mural]

what's the context

[light horizon]

bruh

[solar hawk]

Well, just that

[fair mural]

so context isn't key

[light horizon]

do you have any idea how weird that is out of context

amukh

[solar hawk]

See quwu

There is context

Ur just missing it

[fair mural]

.

[solar hawk]

L bozo moment L

[fair mural]

you answered as if there was no context

[solar hawk]

Ok and why did you believe me

[light horizon]

the context is that amukh showers so little its actually disturbing

[solar hawk]

BRO I SHOWER

IM ABOUT TO RN

[light horizon]

so he expects praise for actually showering

[solar hawk]

Not praise I’m just telling u so yk

[fair mural]

i'm still up for believing that you're andrei's alt for trolling

[solar hawk]

THAT I’m a clean frog

[light horizon]

real

[neat lintel]

[solar hawk]

Can I get skin cancer from not showering

[neat lintel]

[solar hawk]

Hypothetically

[fair mural]

is that your only concern or

[solar hawk]

Like in the extreme case

[solar hawk]

Well

I’m clean so yea

[fair mural]

oh my god just shower

[sonic field]

HUH?

[fair mural]

you're not clean if you don't shower for a week

[solar hawk]

I have waited longer but that is irrelavent

[fair mural]

i'm trying to do my homework but amukh is just too entertaining

[solar hawk]

I just wanna know

If u can die

[fair mural]

you wanna know if you can die from not showering

seriously

[sonic field]

bro

where did you get this from?

[solar hawk]

Today in bio

My teacher said we can die from skin cancer or smthn

[sonic field]

... did they tell you WHY THAT'S TRUE?

[sonic field]

-amukh not payin attention in bio-

[solar hawk]

Grownups commit meiosis, body cells do mitosis

Done bio finished

100% passing grade

[sonic field]

I guess mitosis is gonna give you cancer too huh?

[fair mural]

shoot wrong thing

almost had it

[sonic field]

also

[fair mural]

like i said, amukh is gonna be learning from stuff like this

[sonic field]

y are u still on the shower thing??

[solar hawk]

We already leaned that shit

[solar hawk]

I’m waiting for it to warm

https://tenor.com/view/toad-king-studios-tadpoles-flag-happy-cute-frog-kingdom-gif-19484446

Alr bue

Bye

[neat lintel]

"Domain Expansion: Cellular Respiration" still gets me lmao

[fair mural]

watching that video hurts

it's so funny it kills me

[sonic field]

"nah, I'd win"

LOL

[fair mural]

wait @native oriole watch this biology edit

[solar hawk]

The reason the mitochondria would win

Is fermentation.

[sonic field]

here we go

[fair mural]

nah i'd win

[solar hawk]

It could do alcohol fermentation, or lactic acid fermentation

Wait fuck

Nvm

I’m stupid

Fuck that bye

https://tenor.com/view/nah-i'd-win-gif-7220010602787901432

[sonic field]

LOL

[ripe needle]

[fair mural]

how is it not biology

i mean it's basic biology

but still

[fervent flame]

thats definitively bio

[ocean harbor]

biochemistry, metabolic processes, molecular genetics, hometasis, and population dynamics

biology gives me a headache

[fair mural]

[ocean harbor]

catFone

[neat lintel]

Metabolism

[latent edge]

the shenanigans of the Krebs cycle

I remember making a category out of the cycle

top tier crankery

[sage prawn]

Hey theere

Any lurkers?

[limber thunder]

wha

[neat lintel]

damn this is fire

[naive gate]

ok so

@crimson nebula almost forgot what we were talking about

uhh

for example integral from 0 to 3 of 1/(x-1) dx

is undefined at x=1

so we were talking about using limits to determine divergence/convergence

[crimson nebula]

so, integrate it

or wait

it has a pole on [0,3]

[crimson nebula]

yeah

[rose dawn]

The integral if it exists is, $$\lim_{\varepsilon \to 0} \int_{0}^{t-\varepsilon} \frac{1}{x-1}dx +\int_{t+\varepsilon}^{3} \frac{1}{x-1} dx$$

[crimson nebula]

piecewise it

[naive gate]

so we choose a value for the limit to approach based on the undefined?

[crimson nebula]

$$\int_{0}^{3}f(t)dt=\int_{0}^{1}f(t)dt+\int_{1}^{3}f(t)dt$$

[fathom swallow]

Cycadellic

[crimson nebula]

hes less lazy than me

[naive gate]

lol

so here's choosing limit for upper and lower bounds?

[crimson nebula]

around the discontinuity

[rose dawn]

and actually

[crimson nebula]

then if it converges, the discontinuity converges

[rose dawn]

those should be the same limit

[crimson nebula]

not necessarily, i dont think

for example, the floor(x) has a convergent area function

yet the one sided limits arent the same

[naive gate]

so what if a function doesn't have a discontinuity?

[crimson nebula]

then mean value theorem holds simply

so you can integrate it normally

[fathom swallow]

Austin

[naive gate]

so once u integrate it normally

then do u do that comparison?

[crimson nebula]

yeah

so, with the example from earlier, it was $$\big\int_{1}^{\infty}{e^{-x}}dx>\big\int_{1}^{\infty}\frac{e^{-x}}{x}dx>0$$

[fathom swallow]

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

[crimson nebula]

its kind of like using the squeeze theorem

[naive gate]

can u also use limits here to prove?

[crimson nebula]

but with a finite distance instead of an infinitesimal distance

[naive gate]

i c

[crimson nebula]

well, by definition $\big\int_{1}^{\infty}f(t)dt=\lim_{x\rightarrow\infty}\big\int_{1}^{x}f(t)dt$

[fathom swallow]

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

[crimson nebula]

then, its easy to apply the fundamental theorems, if mvt

[naive gate]

that's right

ok

[naive gate]

and if i want to use the comparison theorem also?

[crimson nebula]

for this one, you would simply need to show that $\lim_{x\rightarrow\infty}\big\int_{1}^{x}e^{-t}dt$ converges

[fathom swallow]

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

[crimson nebula]

then the convergence of $\lim_{x\rightarrow\infty}Ei(x)$ would follow

[fathom swallow]

Cycadellic

[naive gate]

i recall that it splits into integral from 0 to 1 of e^-x dx + integral of 1 to ∞ of e^-x dx

another way

i mean?

[crimson nebula]

it has a pole at x=0

and the area diverges

[crimson nebula]

if we wanted to talk about Ei(0), we would need to consider -1 to 0, and 0 to 1 seperately, and see if the total converges

turns out, it diverges, though

cause the degree of the denominator is 1

[naive gate]

wdym

[crimson nebula]

consider x=0 on 1/x

this is a pole

[naive gate]

ah

ic

[crimson nebula]

some people also say singularities instead of discontinuities

[naive gate]

never heard that before

interesting

[crimson nebula]

like

black holes are a singularity

[naive gate]

yea

[crimson nebula]

for the same reason

[naive gate]

makes sense

[exotic reef]

Yeah they are discontinuities

[crimson nebula]

i mean more so, their tendancy

x tends to infinity

[naive gate]

so instead of 0 to 1 and 1 to ∞, it would be -1 to 0 and 0 to 1?

[crimson nebula]

einstein made a differential equation, which schwarzchild solved and find a singularity called the swarzchild radius

i think theyre called the einstein field equations, but dont quote me

[leaden surge]

Well no the singularity is at the center

The schwarzchild radius is the point of no return, the event horizon

But the event horizon isn’t a singularity

[crimson nebula]

black holes have a size, they just perpetually collapse

maybe

continuous postulate

no

i mean, this specific solution causes a singularity in density

it shoots off to infinity

[naive gate]

infinite density at the singularity

[crimson nebula]

but it will never become infinitely dense

[leaden surge]

Yeah but you said “the schwarzchild radius is the singularity”, which is wrong. The schwarzchild radius is the distance of the event horizon from the center, and at the center is the singularity

[crimson nebula]

yes, it was semantically incorrect

i misworded

[rocky shuttle]

blackchild radius

[crimson nebula]

this is what i mean concretely

[naive gate]

ok so what about integral from 0 to ∞ of e^-x^2 dx?

could that not use 0 to 1 and 1 to ∞?

[crimson nebula]

yes

then 1 to infinity follows from e^-x>e^-x^2 for x>1

[naive gate]

why can't e^-x work with these?

[crimson nebula]

it wont work on 0 to 1 because e^-x^2 > e^-x for x on [0,1]

so, e^-x is not an upper bound

it is a lower bound, but so is 0

but you can pick a constant like 2 and itll hold

just because we arent dealing with infinity in this integral

[naive gate]

what are the guidelines for picking the bounds?

[crimson nebula]

f and h bound g in some interval when, for all x on that interval, f(x)>g(x)>h(x)

if f and h have finite area, then the theorem is applicable

[naive gate]

i mean can u arbitrarily pick them?

[crimson nebula]

yeah

[split pawn]

Hi disussy

[naive gate]

so why woould one choose 0 to 1 and 1 to ∞ for this one?

[split pawn]

Help channel in disussy

[crimson nebula]

because e^-x was the first thing that came to mind which has a convergent area

the problem was that the inequality didnt hold for 0 to 1

but it does hold for 1 to infinity

[naive gate]

for e^-x^2?

[crimson nebula]

that e^-x > e^-x^2

[crimson nebula]

this is only true for all x > 1, and all x < 0

[naive gate]

ok i think i have too many threads going on. let's start back at the beginning step

so given the integral from 0 to ∞ of e^-x^2 dx, what's the first step for one to show convergence

[crimson nebula]

turn it into a limit

this way, we can use a variable instead of the indeterminable infinity

then we need to find an upper and lower bound

we know 0 is a lower, then i gave the piecewise {2, if x in [0,1); e^-x, if x in [1, infinity)} as the upper bound

[naive gate]

so why 2 here?

[crimson nebula]

arbitrary choice which converged and was greater than the function

in fact, the whole thing is an arbitrary choice which has a converging area and is greater than the function

[naive gate]

greater than because negative exponents puts it on denominator which makes it less than 1?

[crimson nebula]

we could do 20e^-x on 0 to infinity, now that i think about it

1 integral is faster than 2

[naive gate]

so in what scenario would you use 2?

[crimson nebula]

piecewise functions

you integrate by their pieces

[naive gate]

ok well when would u use piecewise?

[crimson nebula]

i think youre over thinking this part a little

we simply care about finding any function which has a converging area, which is an upperbound

[naive gate]

so by any function

u mean

hmm

well any function that is bigger than 1 would work right?

[crimson nebula]

the problem with that is the infinity boundary

a rectangle with height 1 and infinite length would not help us

[naive gate]

why

[crimson nebula]

you would end up showing the upper bound diverges, which is not useful

its strictly
if upperbound area converges, and lower bound area converges, then integral converges

if upperbound area does not converge, or lower bound area does not converge, then we dont know anything

[naive gate]

so both bounds have to converge for the integral to converge and both bounds have to diverge for the integral to diverge?

[crimson nebula]

yeah

[naive gate]

this is not talking about diverging right?

[crimson nebula]

there is a similar theorem for divergence

but the theorem we are using means nothing if the bounds diverge

[naive gate]

why? aren't we trying to find out whether the function diverges or converges?

or is it that if it doesn't converge, it's automatically divergent?

[crimson nebula]

in this specific theorem, think about the boundaries as saying the area cannot be bigger than the area of the upper - area of the lower

you end up saying the area cannot be bigger than infinity if both diverge

which, yeah, its true, but that applies to everything

so that particular theorem is useless in the divergent case

whereas, if they converge to some value L

you say the unsigned area cannot be bigger than L

which is meaningful

[naive gate]

so would this not be a divergent case?

since the integral is to infinity?

[crimson nebula]

the integral of e^-x^2 from -infinity to infinity is sqrt(pi)

i think

yeah

clever solution there

with polar integrals

nothing fancy

[naive gate]

from 0 to ∞

[crimson nebula]

i mean, if it converges on -infinity to infinity, it converges on 0 to infinity

but yeah

[naive gate]

how did you conclude convergence?

[crimson nebula]

its even, so itll turn out to be sqrt(pi)/2

[naive gate]

howd u get that

[crimson nebula]

its a famous integral

the gaussian error function

its a consequence of the conversion between polar and rectangular integrals

i think i remember the proof

[naive gate]

and are u saying u use 2 functions for an upper and lower bound?

[crimson nebula]

not to solve it

this just shows convergence which is what you care about

the specific solution doesnt matter, i was just making a point that it did actually converge

[naive gate]

ok

[crimson nebula]

this is totally irrelevant to the point, but in case youre curious

given $x^2+y^2=r^2$, $\tan^{-1}{\frac{y}{x}=\theta$ $dydx=rdrd\theta$
$$\big\int_{-\infty}^{\infty}e^{-x^2}dx$$

consider
$$\big\int_{-\infty}^{\infty}\big\int_{-\infty}^{\infty}e^{-(x^2+y^2)}dydx$$
$$=\big\int_{-\infty}^{\infty}e^{-x^2}\big\int_{-\infty}^{\infty}e^{y^2)}dydx$$
$$=\big\int_{-\infty}^{\infty}e^{-x^2}dx\big\int_{-\infty}^{\infty}e^{y^2)}dy$$
$$=(\big\int_{-\infty}^{\infty}e^{-x^2}dx)^{2}$$

but
$$\big\int_{0}^{2\pi}\big\int_{0}^{\infty}e^{-(r^2)}rdrd\theta$$
$$=\pi$$
so
$$=(\big\int_{-\infty}^{\infty}e^{-x^2}dx)^{2}=\pi$$

[fathom swallow]

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

[crimson nebula]

but again, totally irrelevant

[crimson nebula]

yea

but 0 is a function with 0 area, so its really useful if we can get f>0

[crimson nebula]

i would stay up longer and explain, but im struggling to stay awake rn, so im going to go to bed

[naive gate]

Haha sry. Im also on the brink of falling asleep. Would you be available tomorrow morning or noon? I’d like to clarify things if u dont mind. U’ve been a huge help

[crimson nebula]

Depends on homework, but i may

Wont wake up until pretty late, though

[naive gate]

Ok i see. Around what time?
And I’ll try to keep the time limited so i don’t spiral down too long lol

[crimson nebula]

Idk tbh, i have a very capricious sleep schedule when i dont have to wake up

[foggy mason]

Does anyone have tips on how to study for their math courses? For reference I took pure math courses in my first year and while I really enjoyed them, I thought the difficulty was a bit much for me and I also wasn't really planning to go into mathematical research, so I dropped down to slightly less pure mathy courses in second year. Still very much a proofs based course but they spend a bit more time getting you to understand definitions and theorems and gain some intuition. I thought that maybe I'd do better this time but I really haven't. A lot of the questions on the test are multiple choice and true or false questions (e.g determining what the tangent plane is of a particular set at a particular point) which I keep bottling. My theory was that I didn't develop the skill in first year because the pure math courses pretty much had only proofs on the tests. But it feels like a dumb excuse because coming up with examples and counterexamples is an essential skill regardless, and it's not like I'm doing great on the problem sets either and those are supposed to be all proofs. We get weekly worksheets so I pretty much did as many as I could to study for the test in preparation for those objective type questions but somehow I did worse than before. I'm not really sure where to go from here

The course is a multivariable calculus course. My other math course is combinatorics which seems to require a different skillset altogether so I guess its a little different for that one

[restive field]

anybody online?

[neat lintel]

no

[narrow hamlet]

I have a question. Let's say that you are correct and the problem is the lack of understanding from your first year. Is there a way to remedy this? I'm just wondering because you say it's a dumb excuse, but if it is a possibility then it should probably be explored a little bit. Just an idea.

[karmic sandal]

Why do you think it can be not as hard as it is?

[foggy mason]

I dont think its lack of understanding, I meant I don't really have the skill of looking at multiple options and determining what is correct, since we never really had tests like that in first year. I thought it was a dumb excuse because I was still taught to think of examples and counterexamples to results and so just saying that I'm not used to multiple choice and true or false questions is a poor excuse :)

[foggy mason]

That is a fair question. I guess it comes down to feeling like I've had a year under my belt to make mistakes in some really difficult first year courses, and now is my chance to apply those lessons in a less intense setting

[karmic sandal]

Learning grammar and writing effectively are two very different skills, learning definitions and theorems is a different skill than applying them.

I'm not sure there's a way around it being difficult and it's hard to say what you could be doing differently to make it not harder than it needs to be. The first time I really struggled with something similar was in a Discreet Math course and I found that the best way through was to find many different problems to think about. It didn't really matter if I was able to prove or solve anything on my own (I asked other students and the professor for help often) but I was able to think more clearly on exams as a result.

[foggy mason]

Thank you so much! I'll keep that in mind

[livid spoke]

I am here to rant

[limber depot]

hi