help-49

[leaden seal]

when i put the limits in

[dawn dagger]

you forgor the minus

it's -2/1

[leaden seal]

is it

[dawn dagger]

minus distributes

-(a+b)=-a-b not -a+b

[leaden seal]

got it

[dawn dagger]

yay congratz

[leaden seal]

how would you apply the limits here

would i put it where it says 2a+1

[dawn dagger]

u apply the limits only for the variable u integrated with which is x

a remains, because it's a constant

[leaden seal]

i didnt see the x

,w integrate dx

[grand pond]

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[dawn dagger]

e^(2a+1) is a constant

[leaden seal]

so

are we integrating 1 or dx?

[dawn dagger]

1

dx is just an "indicator" what variable you integrate right now

[dawn dagger]

and then u apply the power rule as usual

[dawn dagger]

what is your question

[leaden seal]

is the 6k^2 meant to ne 12k^2

[dawn dagger]

why do you have 6

[leaden seal]

idk

[dawn dagger]

oh i see

[dawn dagger]

yeah 6 remains

[leaden seal]

would you just expand and then square

[dawn dagger]

you get only 6 because the 2 cancels

[leaden seal]

so 6k^2

,w expand 3(2k)^2/2

[grand pond]

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[dawn dagger]

yea

[leaden seal]

and then

[dawn dagger]

idk what then

[dawn dagger]

ok yea

[leaden seal]

they've just written the decimal as 9/2

[dawn dagger]

yes

thats allowed

and often recommended

[leaden seal]

question: what are they doing where the question makr is?

[dawn dagger]

they are making use of symmetry

[leaden seal]

of the graph?

[dawn dagger]

yes but that seems the wrong way

y is odd not even

so the integral just vanishes

someone thought the function was even and so they can integrate over half the interval but double the area, i guess?

[leaden seal]

how would i know about the symmery without plotting the graph?

[dawn dagger]

by definition

[dawn dagger]

you show this property holds then it's odd

[dawn dagger]

so the check:

f(x)=x(x²-9)

f(-x) = (-x)((-x)²-9) = -x(x²-9) = -f(x)

[leaden seal]

yh i know this

[dawn dagger]

yea and this is also not only necessary to know but also sufficient

[leaden seal]

ididnt know the graph would be symmetric tho

[dawn dagger]

you dont need a graph

o

[leaden seal]

so

[dawn dagger]

sometimes when the bounds are equidistant then it's worth to see first if the function is odd/even

[leaden seal]

is f is odd then the graph is symmetric?

[dawn dagger]

point symmetric

not symmetric regarding the y-axis

like x² for example

[leaden seal]

about the origin

[dawn dagger]

yes

good mention

[grim heath]

hi, universe

nice to meet you

[leaden seal]

so if f(x) is odd then its symmetric about the origin?

[grim heath]

Mr. Zzzzz

[dawn dagger]

yes

[grim heath]

you are wrong

mistack

mistake

[leaden seal]

alr ty

[grim heath]

sendme DM

i will explain

okay?

i am waiting u

[dawn dagger]

if u r done u may close this channel

[grim heath]

hello

[leaden seal]

im not

[dawn dagger]

o oki

[leaden seal]

i'll open another one tho

[dawn dagger]

sure

[leaden seal]

.close

[midnight plank]

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[grim heath]

sure

[midnight plank]

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[leaden seal]

how would you factorise this

[leaden seal]

for this

a cubic factorial

[ocean wren]

i cant help with the integrating part but fpr the factoring part

try to factor between x^3 - 2x^2 and -x+2

you should end up with the same brackets.

[leaden seal]

hmmm

but its cubic

[prime hornet]

you can factor by grouping

that's what Cheze is getting at

[ocean wren]

you can factor out the x^2 from the x^3 -3x^2.

[prime hornet]

alternatively, you can use something like the rational roots theorem

[ocean wren]

x^3

fu

[leaden seal]

how did you get -2x^2

[prime hornet]

it's in the cubic

[ocean wren]

its in the cubic

x^3 - 2x^2 - x + 2

[prime hornet]

Cheze is suggesting you group the cubic like (x^3 - 2x^2) - (x - 2) and factor each term separately

[leaden seal]

the cubic is x^3 - 2x -x + 2

[prime hornet]

you're missing the square on the 2x

[ocean wren]

it says 2x^2 on it

[ocean wren]

this

[leaden seal]

OH

[ocean wren]

have you by chance read it wrong and it being the only reason yuo cant factor it?

[leaden seal]

i meant this

[ocean wren]

it is still 2x^2

[leaden seal]

oh yh

i've written it wrong mb

[ocean wren]

ok so you just misread the qeustion?

yeah

[leaden seal]

i still need help facotring tho

so you're saying group terms?

[ocean wren]

ye

first two and the last two

take out the HCF between them

[leaden seal]

and then factorise them?

separately?

[ocean wren]

then the brackets should be the safe.

same*

[ocean wren]

ye

first two terms seperately

and the other two seperately

[leaden seal]

so

like h

(x^3-2x^2)

[ocean wren]

for eg. x^2 + 2x + x + 20
-> x(x+2) + 1 (x+20)

[ocean wren]

yeah

like that

[leaden seal]

so

that would be x^2(x-2)

[ocean wren]

yeah

[leaden seal]

and hen?

[ocean wren]

now do the same thing to the last two numbers

-x + 2

[leaden seal]

wdym

[ocean wren]

what is the HCF of x and 2?

i should say -x and +2 for that matter but ye

[leaden seal]

what am i factorising

[ocean wren]

factor out something common from -x abd +2

[leaden seal]

the last term is only -x+2

[ocean wren]

yes it can still be factored

[leaden seal]

oh so 1(-x+2)

[ocean wren]

yeah but

since the x is negative

you can factor out a -1

instead of a 1

so what does the bracket inside become if i take out a -1 instead of a 1?

[leaden seal]

-1(x-2)

[ocean wren]

yeah so it become

x^2(x-2) -1(x-2)

you see how the brackets are the same?

[leaden seal]

yh

[ocean wren]

now if i treat x^2(x-2) and -1(x-2)
as numbers in themselves

you see how they have a common HCF of (x-2)?

[leaden seal]

yh

[ocean wren]

so i can just

(x-2)(x^2 - 1)

because i took out common from x^2(x-2) and -1(x-2)

which is (x-2)

and the thing left is (x^2 - 1)

which is the last bracket

[leaden seal]

hmm

[ocean wren]

now if you want to take it a step furthur and factorise the x^2 - 1 bracket then do it but i dont know if it will help in integration because i havent even learnt basic trig yet

😭

im not doing calculus.

[leaden seal]

yh ok

so just factorise x^-1

[ocean wren]

yeah. If it helps that is.

[leaden seal]

it does

[ocean wren]

yeah so the cubic should boil down to that

damn what the fu

[leaden seal]

becus we need this

im still confused about

[ocean wren]

integrating part

?

welp anyways. im going now. Hope my part helped with ur factoring

[leaden seal]

ah okay

it did

thanks

[midnight plank]

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[buoyant fjord]

how hard is intro analysis 2 compared to intro analysis 1

[junior flower]

hard to say. do you have a description of each?

[buoyant fjord]

yea but they are oudated

cuz

the analysis 1 description says we cover integration

but we ended with rudin chapter 5

differentiation

its prob a few more chapters of rudin

the general metric space stuff was genuinely cooking me a little bit

is that left behind

[junior flower]

no two courses are the same

[sick fossil]

So do you have a list of the topics covered? That would help

e.g. analysis 1: sequences, series, continuity
analysis 2: lebesgue spaces, locally convex spaces

[buoyant fjord]

maybe i should js email the prof

[buoyant fjord]

yeah lemme see but its oudated

Convergence of sequences and series of functions, differentiation and integration in higher dimensional settings. Inverse and implicit function theorems.

[sick fossil]

how about from your own recollection of what they said or what problems you had to do

[sick fossil]

this is analysis 1?

[buoyant fjord]

analysis 2

i just finished analysis 1

[buoyant fjord]

no one told me anything and my prof didnt know

[sick fossil]

but you know which problems you had to complete and hand in, right? So you could reconstruct a list of topics from that

[buoyant fjord]

from analysis 1 yes

[sick fossil]

then intro analysis 2, from your description, seems more annoying than intro analysis 1 if you aren’t using the Lebesgue integral

[buoyant fjord]

basically we just did metric spaces and compactness, sequences (convergence, cauchy, upper/lower limits,), limits, continuity (just from R into R really with mostly the standard metric, differentiation (mvt, lopital, taylor polynomials)

no series tho

[sick fossil]

so yeah it sounds like it still

so you haven’t done the Lebesgue integral

[buoyant fjord]

no idk what it is

just that its more general

[sick fossil]

I don’t understand why people teach the higher dimensional Riemann integral

and they have to introduce “content zero”, whatever that is

[buoyant fjord]

i dont have to take analysis 2 i was just deciding between a few options

[buoyant fjord]

is it not good

[sick fossil]

It seems like you could just teach higher dimensional integration after the Lebesgue integral, but take my opinion with a grain of salt because I haven’t learned the higher dimensional Riemann integral

the Lebesgue integral is general enough to cover higher dimensions after you define product measure spaces, which you should do anyway

[buoyant fjord]

alr thanks for your time

.close

[midnight plank]

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[buoyant fjord]

one more thing. if i wanted to learn some of this on my own what book could be good

?

[sick fossil]

to me, the “integration in higher dimensional settings” seems like the annoying and unnecessary part of the course if you’re doing that rigorousl

everything else is important

[sick fossil]

hmm

[sick fossil]

I’m also going to try to learn these topics, and so far it seems like for differentiation in higher dimensions and the inverse and implicit function theorems, Folland’s Advanced Calculus and Boothby’s book Intro to manifolds (or something) could be good

[buoyant fjord]

ok ill check those out when im done covering the parts we skipped, thanks again

.close

[sick fossil]

You should ask in #book-recommendations , and ping me if you do, because I also want to know good books for these

In case the two books I mentioned aren’t the best

[midnight plank]

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[last slate]

Hi

[last slate]

I’m trying to find out how to find the range for this matrix A that defines a linear transformation

The book didn’t even explain how to find the range except show one example where they took the RREF of the matrix and used its row vectors as a basis for the range

Which also i didn’t understand

But it makes more sense to be the column space

Not row space

Oops

Never mind the book took the column space

Not row space

Dude 😂

My bad

But do we need to do RREF??

[thorn tapir]

it's easy ¯_(ツ)_/¯

you need to find a basis somehow

[last slate]

Is it because RREF ⇒ independent vectors and we need independent vectors because basis ⇒ independent vectors ?

[thorn tapir]

^^

it's a computationally efficient way of just checking if a vector is LI from a set of others

it might be the most computationally efficient, even

[midnight plank]

@last slate Has your question been resolved?

[sick fossil]

it’s not really that

[sick fossil]

it’s just that RREF only requires calculators and not a brain, so it can be useful to reduce all problems to taking RREF using the right theorems

[midnight plank]

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[azure bolt]

not sure how to manipulate the end

[lyric charm]

this is a geometric progression in all but name

anyway, you may have better luck showing that

(1-r)(1+r+r^2+...+r^n) = 1 - r^(n+1)

[azure bolt]

,,\begin{align*} a + ar+ar^2 + \cdots + ar^k + ar^{k+1} &= \frac{a(r^{k+1}-1)}{r-1} +ar^{k+1} \ &= \frac{a(r^{k+1}-1)}{r-1} +\frac{(r-1)(ar^{k+1})}{(r-1)} \ &= \frac{ar^{k+1} -a +(r-1)(ar^{k+1})}{(r-1)} \&= \frac{a(r^{k+1} -1 +1)(r-1)(r^{k+1})}{r-1}\end{align*}

[lyric charm]

ah, so we are doing it by induction, are we.

or. wait. hold on.

(3) looks sussy.

[azure bolt]

did i factorise wrong

[lyric charm]

yeah you messed up the numerator p bad.

[azure bolt]

what is it with my factorising today

[lyric charm]

factor out just a for a start

[azure bolt]

Is this right?

[lyric charm]

well it's correct but you haven't done what i said to do yet

[azure bolt]

factor a out

hmm

do you mean the ar^{k+1} term?

[lyric charm]

no

i mean the letter a

[azure bolt]

just a?

[lyric charm]

factor it out of everything present

yes just a

[grand pond]

calvin

[azure bolt]

is this right?

[lyric charm]

no you are still messing up bad and probably overthinking it

[azure bolt]

aww

sorry

why dont i just factor out ar^{k+1}

[lyric charm]

$a(r^{k+1}-1)+ar^{k+1}(r-1) \ = a[r^{k+1}-1+r^{k+1}(r-1)]$

[grand pond]

Ann

[lyric charm]

this is what i wanted you to do on the numerator

this and naught more

$a(r^{k+1}-1)+ar^{k+1}(r-1) \ = a[r^{k+1}-1+r^{k+1}(r-1)] \ = a[r^{k+1}-1+r^{k+2}-r^{k+1}]$

[azure bolt]

oh what

[grand pond]

Ann

[azure bolt]

im stupid

[visual elk]

never mind a small question

must we do it with induction?

[azure bolt]

Well

The example was induction

[lyric charm]

there's not much of a way around it unless you're willing to cause a wind storm with your hand waving

[azure bolt]

uh

idk

[lyric charm]

was responding to ishmam

[azure bolt]

oh

Whats next

[lyric charm]

you got them r^(k+1) terms about to annihilate each other

[azure bolt]

wym

i cancelled them

im left with

,,\frac{a(r^{k+2}-1)}{r-1}

is this normal undergrad stuff

[grand pond]

calvin

[azure bolt]

because i was just reading a number theory book

And proofs came up

what do i do from here

[visual elk]

Is this proof sound?

[midnight plank]

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[last slate]

Elo

[pearl hull]

yes

[last slate]

Is this just a fancy wording for the gram Schmidt process ?

[last slate]

Interesting name

[last slate]

🤔

[pearl hull]

never heard of that before, can you briefly explain the context?

[last slate]

The gram Schmidt process is basically taking a basis and make it orthonormal through a process of using projection vectors

[pearl hull]

I just googled it, looks like you can apply that in this question

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

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[stuck willow]

What is 67+69

[static moth]

136

[pearl hull]

.solved

[midnight plank]

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[pearl hull]

#chill if you wanna troll or smth

[midnight plank]

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[last slate]

Hi can I get quick help

[last slate]

What do they mean by the projection on the span

<@&286206848099549185>

[latent wadi]

(1, 0,-2) and (2, -1, 1) span a plane right?

Then you need to find the vector u, projected onto this plane

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@last slate Has your question been resolved?

[last slate]

Yes

Like the distance

[midnight plank]

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[sturdy kindle]

How to solve this without delta method? Factor?

[inner barn]

what is delta method

[lethal path]

it's not possible without the quadratic formula

[sturdy kindle]

Look it

[lethal path]

well, you could complete the square but that's the same method

[sturdy kindle]

But the assignment does

Use quadriceps formula or not

[lethal path]

have you seen this method before?

[sturdy kindle]

But otherwise the other method would be factoring. @lethal path

[lethal path]

it's not factorisable

[sturdy kindle]

No wasnt taught

[sturdy kindle]

What condition can be factor able

[lethal path]

that the discriminant $\Delta$ is a perfect square number

[grand pond]

south

[sturdy kindle]

This is a bullshit

1 - 8/3

Is the delta which can’t

[lethal path]

yes, in fact the solutions are complex numbers

they have i

[sturdy kindle]

Okay so this number isn’t in real numbers

[lethal path]

exactly

[sturdy kindle]

x /€/ R

[lethal path]

$x \notin \mathbb R$

[grand pond]

south

[sturdy kindle]

Yea it’s the same

Yep

.close

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[late rover]

I can't wrap my head around this solution

What sum of products do they mean here and why they're corresponding to the words of length n

Ping on reply please

[runic hamlet]

if we multiply out eg (a+b+c+d)(a+b+c+d) we get aa+ab+ac+ad+ba+bb+bc+bd+...

so we get all words of length 2 in the letters a,b,c,d

similarly for (a+b+c+d)^n

@late rover

[late rover]

Oh wait yeah, I did thought about that but somehow I got lost

Wait letmme try to digest this solution

omg I get it now, what a brilliant

hang on I still want to make sure I don't misunderstand anything

[midnight plank]

@late rover Has your question been resolved?

[late rover]

I can ensure that I understand it fully now as I has written it down w/o looking at the solution

Thanks y'all

[midnight plank]

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[late rover]

[midnight plank]

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[tame summit]

Let's be honest, the question's more scared of me then I am of it but wth. How do I...what am I even supposed to do here? It's like my brain is lagging??

[tame summit]

(also, sry, I accidentally sent the pics in the wrong order)

(this is why I'm incompetent or smth)

[grand gust]

I suppose the price on the graph is quoted per quintal?

[tame summit]

Yeah

[next bobcat]

Do you need help with all the stuff

[grand gust]

so which part(s) are you confused by?

[tawdry pebble]

Isn’t this categorized under economics type of help? Or still allowed? Just asking just in case i might need help in economics things

[grand gust]

well it's kind of math-adjacent I suppose, but the mods can determine if need be

[late rover]

you can ask for anything actually, at least it's science related like that one time a girl ask about biology meiosis process in help channel

[lethal path]

we get people asking about stoichometry in chem here too

[lethal path]

okay that's over the line

nothing to do with maths

[late rover]

-# But she did got help lol

[grand gust]

there's people asking biology here?

damn, okay

[tame summit]

Amazing

[orchid fossil]

i asked for ochem twice or thrice btw

physics is very rare idk why

[grand gust]

physics ain't rare from my experience but I'm not discussing this in OP's channel

[tame summit]

Anyways, I'm losing it cause I have an entrance in like 2 days and my god am I scared, idek what to look for in this?? How do I study for this??

[grand gust]

this looks like you need to know how to interpret the graph, and separately find the revenue per acre of both products.

(not profit, revenue. you can find the profit afterwards)

the idea is that given the yield per acre and the price per quintal, you can know how much money you'll earn from planting an acre of each product and selling all of the yield

then, you can subtract the cost of planting those products from the revenue to get your profit, and thus your ROI

a simple example

suppose an acre of paddy has a yield of 2kg, and the selling price per kg of paddy is $20.

that means, for every acre of paddy I plant, if I sell every last grain I get, I'll have $40

now, this is only the revenue, because there are costs involved in planting the paddy as well, of course.
so say that this cost is $15 per acre of paddy.
that means that for every acre I plant, I first spend $15 to plant them, then I get $40 of revenue, so my profit is $40 - $15 = $25, and my ROI is $25/$15 = 1.67

wtf

[tawdry pebble]

is this about the elasticity curves demand and supply?

[tawdry pebble]

alright thanks

[grand gust]

this concept translates directly into your graph

except that you have to be careful, because the selling price of both products increase over time, and therefore the profits also increase over time.
if asked to calculate a profit, ensure you are looking at the selling price of the right product and the right time.

also, the yields per acre of each plant is different, so there's that to consider.
however, you can precalculate all of this, so not too big of a deal.

[tame summit]

Hmmm, I see

Thank you for explaining, I'll give it another try

Wait uh, also, question number 45

What

(no, that's it. That's literally my question: what)

[grand gust]

let me have a look at that

I see, so there's now an inventory cost

well the inventory cost I presume is a simple annual rate, so at 12% per annum and a holding period of 6 months (aka half a year), the effective rate charged would be half the inventory cost
so factor in the inventory cost, which I presume is based off the total item value

[midnight plank]

@tame summit Has your question been resolved?

[tame summit]

I'm guessing I have to subtract inventory cost from the total thing?

Or well no sorry

Subtract it from profit if I have to find a net profit

[grand gust]

subtract it from the revenue separately, or add it to the cost to be subtracted from the revenue together

either way works and is the same

[tame summit]

Okay, I see, thanks

[grand gust]

nps

[midnight plank]

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[slow thorn]

why not 16? you would get T to be proportional to a^3/m

[slow thorn]

came here cause it was more of a calculus and proportionality question than physics

most of the solutions just assumed the masses to be with constant acceleration and got t^2 to be proportional to a^3/m

[midnight plank]

@slow thorn Has your question been resolved?

[orchid fossil]

take hints from the standard irodov qn here (u can find its soln in hcv btw)

[slow thorn]

not the same question bro

i get what you're trying to say

but the particles are accelerating here

and the velocity depends on the side length of the triangle

i did try solving that way and got the relation that T is proportional to a^3/m

like i stated at the start

actually wait i think i made a mistake

oh godddddd

thanks for the support thoug

.close

[midnight plank]

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[rocky quest]

am i interpreting this right

[midnight plank]

@rocky quest Has your question been resolved?

[rocky quest]

.close

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[leaden seal]

is iBP being done multiple times here?

why?

[lyric charm]

do you mean why like "why do we do it like this"?

[leaden seal]

yes

[lyric charm]

well watch what happens to the shit inside the integral with every ibp application

the exponent on the x goes down

eventually to 0

and then it is just a raw sine or cosine term

[leaden seal]

oh

yh okay

[grand pond]

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[leaden seal]

,w integrate cosx

[grand pond]

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[leaden seal]

wouldnt the 4x^3 be negative

[midnight plank]

@leaden seal Has your question been resolved?

[grim vector]

Integration of -sin give cos

The minus is actually with the sin not the 4x³

[midnight plank]

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[leaden seal]

anyone know why my answers are different?

[twilit jetty]

what kind of website or software is that

[leaden seal]

desmos

calculator

[twilit jetty]

how did you get this to the right of the calculator

[leaden seal]

oh thats just a document

[twilit jetty]

did you put in x = 0 yet

you put in x = 1 so far

[leaden seal]

i thought x=0 would just give 0

[twilit jetty]

it doesnt

you should try it with the calculator

[leaden seal]

ok

so iys -18 for 0

[twilit jetty]

iys?

[twilit jetty]

ans - (-18) instead

[leaden seal]

=0.0804

,w integrate du

[grand pond]

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[leaden seal]

,w integrate dx

[grand pond]

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[twilit jetty]

what are you doing

[leaden seal]

just checking smth

[twilit jetty]

.close if you have another question, you can ask it in a different channel

[midnight plank]

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[twilit jetty]

G is finite. Let p be the smallest prime dividing |G|. Let H ≤ G of order p. Prove H ≤ Z(G).

[visual tiger]

!status

[midnight plank]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[twilit jetty]

1, I can be sure this time that I truly have nothing

[visual tiger]

ok, so if p divides |G|, give me a list of things we can say

about the elements and or subgroups of G

[twilit jetty]

for one I found a typo in the question I just sent, Ill change that

I know the H in the edited question exists

it would have to be the smallest nontrivial subgroup of G, maybe

its cyclic

nothing new has come up since then

[visual tiger]

ok

so you want to show elements of H are in the center of G

Do you know any characterizations of Z(G)?

[twilit jetty]

whats a characterization?

you mean like a definition?

[visual tiger]

well more than a definition

can we write it, for example, using a ||homomorphism||?

[twilit jetty]

I couldnt find one

[visual tiger]

Ok, here's a hint

If h is such that gh = hg

Then ghg^-1 = h

Do you know a bit about group actions?

[twilit jetty]

I couldnt have remembered anything about the derived group on the exam, so lets go with no

so far two of the 3 questions Ive been able to find an answer for with an existing trick Ive remembered from for example the homework

[visual tiger]

Does Fixators, orbits and/or stabilizers ring a bell?

[twilit jetty]

dont know what a fixator is, the others Ive heard of them but dont know much about them

for one of the homework assignments we had to do quite a bit about the derived group, dont really know much about it though

[visual tiger]

Ok

Let's try another approach

Do you know about automorphisms and such

[twilit jetty]

lets go with sure

[visual tiger]

Ok

So, if, for $g\in G$, I define the automorphism $\phi_g(x) = gxg^{-1}$ on $G$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

We can then look at the morphism that sends g to phi_g

And we notice that this morphism has kernel = Z(G)

Mmh let me think

[twilit jetty]

if its any idea, ^p is the smallest power that turns elements into identities

but I cant ever get the elements of <q> or whatever Im calling it to stack up conveniently

heres something unrelated but interesting

5 of the questions I was able to answer during the exam involved stuff directly relating to the textbook or previous exam questions

that left 3 I couldnt answer, of those, 2 were answered using tricks that are both outside of the textbook

so if its any indication, I cant just do something textbook for this problem

to be honest all Ive been doing is raising things to ^p powers and then seeing they dont work

[visual tiger]

wait do you have sylow theorems

[twilit jetty]

no

wouldnt remember them if I did

[visual tiger]

yeah ok

[twilit jetty]

we got normal towers though

G isnt given to be solvable though

and theres no guarantee the first nontrivial group in that tower is <q>

let along whether Z(G) is in that tower at all, it could be a normal subgroup that just gets left out

[visual tiger]

ok, do you know about the normalizer

[twilit jetty]

I heard of it once, does that count

[visual tiger]

could you tell me what N_G(H) is then

because we might need it

[twilit jetty]

elements n where nHn^-1 = H
follows that its the largest subgroup of G that H is a normal subgroup of

it might have been given during the exam that H is a normal subgroup, but I cant recall

we could just include that in the problem for convenience, but just to be sure we'll need to prove it

[visual tiger]

yeah, if H is not supposed normal, it's very hard

I'm pretty sure if it's not we can come up with counterexamples

[twilit jetty]

oh wait it most likely was

[visual tiger]

like S_3

yeah take H = <(1 2)>

definitely not in the center

[twilit jetty]

oh yes that, I did find a counterexample during the exam that not all cyclic groups are normal subgroups of G

[visual tiger]

So, if we suppose that H is normal

[twilit jetty]

and we'll have to, really

[visual tiger]

then let's define $\varphi_g (h) = ghg^{-1}$ for $h\in H$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

$\varphi_g$ is an automorphism of $H$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

(you might have seen written $\varphi_g \in Aut(H)$

[grand pond]

Raphaelisius Maximus MMIII

[twilit jetty]

I have, in a different question

[visual tiger]

ok

so that means we can consider $\varphi: \begin{cases}G\to Aut(H)\ g\mapsto \varphi_g\end{cases}$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

what do you think this map is gonna be

[twilit jetty]

as in?

[visual tiger]

what type of map is phi

what properties does it have

[twilit jetty]

I dont believe its isomorphic?

even though nothing about it says otherwise

[visual tiger]

definitely not an isomorphism

[twilit jetty]

oh alr

[visual tiger]

start smaller maybe?

[twilit jetty]

its not injective?

thats rather small to be honest

[visual tiger]

no, but like weaker than isomorphism

[twilit jetty]

homomorphism

[visual tiger]

yes

that's important

(you'll check that it is one)

[twilit jetty]

that I can do

[visual tiger]

so yeah it isn't injective

in fact, what's the kernel?

[twilit jetty]

woah

thats incredible

its H and its also Z(G)

[visual tiger]

the kernel is not H, though h is part of the kernel

[twilit jetty]

oh wait that doesnt do much for us, nvm

I meant to say H and Z(G) are in the kernel

kernel would be the elements where gh = hg

[visual tiger]

ok, so we have that the kernel contains both H and Z(G)

we really want to find what the kernel is, but the best we can say for now is that it's the centralizer of H

$C_G(H) = {g\in G:; \forall h\in H, ,gh = hg}$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

because the elements in the kernel are sent to the identity of H

[twilit jetty]

is it possible that Z(G) = C_G(H)?

[visual tiger]

that would be a juicy finding, wouldn't it?

if we had that, then using that H is in the kernel

then H is in Z(G)

[twilit jetty]

it would if it wasnt that ghg^-1 = h could allow for other not-h elements not to do the smae

[visual tiger]

yep

For now, we won't be able to go further on that lead

oh also wait

[visual tiger]

I don't think Z(G) = C_G(H) actually

C_G(H) is probably much bigger

we'll see why in a moment (I myself can also get confused xd)

For now, we won't be able to go further talking about the kernel

so what are we gonna talk about now?

[twilit jetty]

did raising things to the ^p power repeatedly ever get anywhere?

[visual tiger]

I don't think so 😭

[twilit jetty]

no way

[visual tiger]

let's go back to our lead of $\varphi$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

We've investigated its kernel

what are we gonna investigate now

[twilit jetty]

the Aut(H) version right

Idk its surjective

[visual tiger]

yes this version

[visual tiger]

maybe, maybe not...

The image of phi is, in any case, a subgroup of Aut(H)

what is Aut(H)?

[twilit jetty]

set of isomorphisms from H to itself
Aut(H) is isomorphic to Z/pZ

[visual tiger]

aaaaaaaa almost

H is isomorphic to Z/pZ

[twilit jetty]

oh right H and Aut(H) arent the same thing

[visual tiger]

well now that we know that H is isomorphic to Z/pZ

Aut(H) is isomorphic to Aut(Z/pZ)

no big surprise out of that?

[twilit jetty]

thats not even a theorem we have

[visual tiger]

I don't think it even deserves a theorem

if you have a morphism H -> H
you can create a morphism Z/pZ -> Z/pZ

and vice versa

[twilit jetty]

oh alr

[visual tiger]

like, if you have a morphism f: H -> H

and an isomorphism g:H -> Z/pZ

then you can create a morphism g o f o g^-1

that goes from Z/pZ to itself

and vice versa

[twilit jetty]

that makes sense

[visual tiger]

and if f is bijective

so is the newly created morphism

[visual tiger]

so this

[visual tiger]

any more questions on that assertion?

[twilit jetty]

nope

[visual tiger]

ok

now, we want the automorphisms of Z/pZ

what do we need

if stuck, hint: ||to know a morphism f: Z/pZ -> Z/pZ, how many f(k) do we need to know?||

[visual tiger]

Solution to this question if still stuck: ||f(k) = kf(1), so we only need f(1)||

[twilit jetty]

oh this is no good

question 3 on the exam says it should be (Z/pZ)^x

but testing says it should be (Z/pZ) instead

having a hard time keeping track of which is what

[visual tiger]

well... Aut(Z/pZ) is isomorphic to (Z/pZ)^x

[twilit jetty]

oh right

[visual tiger]

there are p morphisms, because p possible values of f(1)

[twilit jetty]

so theres p - 1 possible values of f(1)

[visual tiger]

only p-1 yield isomorphisms, because no f(1) = 0

so

[twilit jetty]

then is (Z/pZ)^x isomorphic to Z/(p-1)Z?

[visual tiger]

but wait

[unkempt sluice]

?

Yes

[visual tiger]

yes frick

it is cyclic

I mistook with Z/nZ ^x

[unkempt sluice]

Are you still discussing the original question

[visual tiger]

but it's irrelevant

we won't use that (Z/pZ)^x is cyclic

all we need is its order, p-1

so, by morphism theorem

|G| = |Ker(phi)| |Im(phi)|

[twilit jetty]

Ive never seen that before

we did see |G| = |ker(phi)| [G : ker(phi)] before so we can go with that

[visual tiger]

ok let's use that then

[G : ker(phi)] = |Im(phi)| btw

did you never make the connection?

[twilit jetty]

we always called it [G : ker(phi)]

we've never seen |Im(phi)| as written

[visual tiger]

Im(phi) is the image of the morphism phi

|Im(phi)| is its order

the index of G with respect to ker(phi) is precisely how many different images there are

to see that, just notice that $\varphi$ defines an isomorphism between $G/Ker(\varphi)$ and $Im(\varphi)$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

$\tilde \varphi (\tilde g) = \varphi(g)$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

idk how you write equivalence classes

[twilit jetty]

a variety of ways

\overline or as a left coset

[visual tiger]

ok

[twilit jetty]

or with \pi

[visual tiger]

you can define a homomorphism $\tilde \varphi$ such that $\tilde \varphi(\overline g) = \varphi(g)$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

and it's an isomorphism between those

[visual tiger]

in any case, we use this result

notice that |Im(phi)| divides p-1

but it also divides |G|

do you see what's gonna happen now?

[twilit jetty]

I dont know what thats good for

I forgot what ker(phi) even was equal to

is ker(phi) = C_G(H)?

[visual tiger]

yeah, it's not important how it's called

all we know it's some potentially big subgroup of G

maybe bigger than Z(G)

[twilit jetty]

well if C_G(H) = G, then gh = hg and so H < Z(G)
if C_G(H) < G, then p is still the smallest prime dividing |C_G(H)| so H < some smaller subgroup

[visual tiger]

that second assertion is not true

at least I don't see how you'll prove it

Though if C_G(H) = G then yes you hit the jackpot

[unkempt sluice]

Are you trying to translate the idea G conjugate acts on H, each orbit has to have length 1 or p, to a more elementary language for him?

[twilit jetty]

what if p is 2?

if thats a spoiler, then unfortunately Im reading it

[visual tiger]

particular case, not really needed but why not check it out

p-1 is then 1

so what is |Im(phi)|

and so what is |ker(phi)|

[twilit jetty]

oh p still divides |ker(pi)| then

[visual tiger]

well, sure, it comes from the fact that H is in the ker(phi)

but it doesn't have a clear impact on the divisors of |Im(phi)|

Let's try to find |Im(phi)| only using those two infos:

notice that |Im(phi)| divides p-1
but it also divides |G|

[twilit jetty]

I thought p - 1 divides |Im(phi)| instead?

[visual tiger]

I don't know how you got that

[twilit jetty]

I cant really recall whats happening anymore to be honest

whats phi?

[visual tiger]

💀

ok, let's take it back

First of, we had $\varphi_g (h) = ghg^{-1}$ for $h\in H$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

To say that $H\subseteq Z(G)$ is the same as saying $\varphi_g = Id_H$ for all $g$

[grand pond]

Raphaelisius Maximus MMIII

[visual tiger]

any questions from those two facts?

[twilit jetty]

no

[visual tiger]

ok

consider the map $\varphi: \begin{cases}G\to Aut(H)\ g\mapsto \varphi_g\end{cases}$

It's a morphism, and we want to prove that $Ker(\varphi) = G$

[grand pond]

Raphaelisius Maximus MMIII

Raphaelisius Maximus MMIII

[visual tiger]

(that way we get H is in the center)

Still ok?

waiting for your confirmation btw

[twilit jetty]

phi(H) = identity
phi(Z(G)) = identity
phi(C_G(H)) = identity
ker(phi) = C_G(H)
phi(G) ≤ G

Aut(H) isomorphic to Z/(p-1)Z since H is prime cyclic
so |Aut(H)| = p - 1

phi is homomorphic,
ker(phi) normal subgroup of G
|G| = |ker(phi)| [G : ker(phi)]
however there are only p - 1 elements of the output of phi
so [G : ker(phi)] | p - 1 and [G : ker(phi)] | G
so [G : ker(phi)] = 1
so |G| = |ker(phi)|
so G = ker(phi)

as such the only phi possible maps everything to the identity
so ghg^-1 = g for all g in G, h in H
so gh = hg

[visual tiger]

yeah that's it

[twilit jetty]

how did you even think to define phi in the first place

it mapped Z(G) and H to the same thing

[visual tiger]

Whenever there's center/commutativity questions, we tend to think in "group actions"

[visual tiger]

I didn't explicitly say it but this "phi" acts on H using elements of the group G

so you have to think about "commutativity <=> conjugation doesn't do anything"

so you look at some conjugation group action

[twilit jetty]

and you had it act on H only, because we only needed H < Z(G) and not G < Z(G)?

[visual tiger]

you definitely won't have G <= Z(G) (unless you wanna show G is abelian)

in group actions, there are things called stabilizers (which group elements' actions do nothing)

there are also orbits (which elements from H you can attain from another by acting on it)

and fixed points (which elements of H don't change with actions)

here, we were looking to show that every element of G is a stabilizer

and we have results that link the cardinals of all of those between each other

it's a big step to make if it was only used to discuss a single exercise

but it's not easy to talk about the intuition without talking about this tool

[twilit jetty]

how much of the intuition only predicts what a tool can do for us?

[hard umbra]

i feel like you can do this fairly low tech

[visual tiger]

I didn't understand that question

[hard umbra]

if H = <a>, then gag^-1 = a^k for some k

and g^o(g) a g^-o(g) = a^(k^o(g)) = a

so k^o(g) = 1 mod p

but o(g) has no common factors with p-1

[visual tiger]

k * o(g)?

[hard umbra]

no

[visual tiger]

wait

no ok

[hard umbra]

so k=1 mod p

[visual tiger]

pretty low tech indeed

[twilit jetty]

where does g^o(g) a g^-o(g) = a^(k^o(g)) come from

[hard umbra]

so gag^-1 = a^k

then you do it again on a^k

g^2 a g^-2 = g(a^k)g^-1 = (gag^-1)^k

etc

[twilit jetty]

(a^k)^k is a^(2k) though not a^(k^2)

[hard umbra]

[twilit jetty]

oh nvm

thanks

.close

[midnight plank]

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[polar abyss]

i need help with this i’ve forgotten alot about polynomial long division

[latent wadi]

well, if -2 is a root of the equation, then (x-(-2)) = (x+2) divides the polynomial

try writing out the setup for the polynomial long division first, then we can continue explaining

[midnight plank]

@polar abyss Has your question been resolved?

[midnight plank]

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[midnight plank]

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[shut canyon]

What does it mean to find spherical area with a non-euclidean method, not using integrals?

[silent dock]

It just means you are working in spherical (curved) geometry rather than flat Euclidean geometry.

[shut canyon]

Thank you @silent dock ⼼

[silent dock]

[shut canyon]

Do you know a method to look up?

[silent dock]

The most classic one is the Girard’s Theorem

[shut canyon]

awesome, thank you so much, I'll look into this

[shut canyon]

.closed

.close

[midnight plank]

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[solemn sandal]

.reopen

[last slate]

$\begin{enumerate}
\item Perform the division in increasing powers of $1 - X^2$ by $1 - 2X \cos \theta + X^2$ to an arbitrary order.
\item Deduce the value of $1 + 2\sum_{k=1}^n \cos k\theta$ for $\theta \not\equiv 0 \pmod{2\pi}$.
\end{enumerate}$

[grand pond]

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

[last slate]

i need help

i don't know where to start from

i tried doing division couple times but i couldn't find any pattern to make a conjecture

[midnight plank]

@last slate Has your question been resolved?

[unkempt sluice]

You can deduce that sum by using geometric progress and e^iθ thing. But no idea what you were talking about in the beginning

[runic dragon]

Isn't there just a formula for $\sum_{k=1}^n\cos(kx)$?

[grand pond]

Nico

[unkempt sluice]

Yeah, geometric progress

[runic dragon]

Right

[midnight plank]

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[last slate]

it's division by increasing powers

[last slate]

yeah but i guess i need to solve it with the results from the first question

[runic dragon]

Isn't this just Bézout's theorem for polynomials?

[last slate]

not really?

no

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

@last slate Has your question been resolved?

[sick fossil]

what is the first question?

[last slate]

Perform the division in increasing powers of $1 - X^2$ by $1 - 2X \cos \theta + X^2$ to an arbitrary order.

[grand pond]

Darky

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

@last slate Has your question been resolved?

[eternal pawn]

are you sure this is the right statement

[last slate]

should be

[eternal pawn]

do you have any example of similar questions

[last slate]

this forexample

in this question

the division of 1 by (1-X)^2 to an arbitrary order came out to

[deep vine]

You're handwriting is very aesthetic but also completely illegible

You should become a math professor

[midnight plank]

@last slate Has your question been resolved?

[last slate]

i write better on paper 💔

i can't write with a mouse

[last slate]

$1 = (1 - x)^2 \left( \sum_{k=0}^{m} (k+1)x^k \right)

• \left( (m+2)x^{m+1} - (m+1)x^{m+2} \right)$

[grand pond]

Darky

[midnight plank]

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[night gyro]

is this right

[night gyro]

then i just substract

[lyric charm]

your work is a bit shoddy

you would subtract -n^2 to figure out the linear part

[night gyro]

yh sorry

[lyric charm]

which is the same thing as adding n^2 actually

[night gyro]

so how do i do it

[lyric charm]

keep going like you were doing

!occupied

[midnight plank]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[lyric charm]

write things in a less "disorganized chicken scratch i thrive in chaos but then wonder why i get nothing right in math" way tbh

[crimson rivet]

Xd

[night gyro]

so i do 5-1=4 then 3-4=-1 then -9-1=-10

[night gyro]

?

[lyric charm]

you have to keep your work neat and organized

you understand that, right?

[night gyro]

yh ok

[night gyro]

is this right

as u said just continue

[lyric charm]

no, this is not right

[lyric charm]

this error came purely from the fact that you wrote your work in a way that you yourself can't read and understand.

[night gyro]

wouldnt 3--4=-7?

[lyric charm]

no, 3 - (-4) = +7 not -7.

big number minus small number is always positive, mate

[night gyro]

i see

then i just find the nth term of that

[night gyro]

thanks

also any tips to keep my working tidy

[lyric charm]

tip #0 is to write on a PIECE OF PAPER

with a PEN

or a PENCIL

[night gyro]

yes ok

[lyric charm]

because writing with your finger or god forbid a mouse over a digital image

is just a road to disaster

tip #1 is to write SLOWLY

[night gyro]

it is a mouse lol

[night gyro]

ok

[lyric charm]

badbad. bad. BAD!!!

mousewriting bad. bad.

tip #2 is to write WORDS to SAY what you're doing\

to yourself AND the grader

[night gyro]

ok nice

more?

especially in non calculator papers my papers are a mess

[lyric charm]

try your best to separate arithmetic workings from your main/planning work

uhhh do questions that you know how to do without focusing on time but with a focus on neatness

[night gyro]

you know sometimes i feel like ik how to do the question but when i start i get stuck

then i just leave the question do others and reread the question which wastes time

[lyric charm]

sorry, not what i meant

i meant that you should practice types of questions that you otherwise know how to do

like not as part of a mock

but just on their own

and then write your work out neatly for those

[night gyro]

oh i see

[night gyro]

how about mock

[lyric charm]

ig something like

• first pass: do the questions you can take to completion there and then. if a question poses doubt, skip it for now
• second pass: go over the questions you skipped but give each one some more time. if you see a route to completing it, then do it; otherwise save it for the third pass.
• third pass and onwards: do the hard questions one at a time

[night gyro]

alr thanks

.close

[midnight plank]

Channel closed

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• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[formal terrace]

15. In a certain right triangle, the sum of the two legs is 10 cm. Determine the smallest possible length of the hypotenuse

[formal terrace]

Im clueless

[grand gust]

hint: pythagoras

also, perhaps drawing a random right triangle to help you visualize the question may help

[formal terrace]

Ye I figured that much out but idk how to proceed

[gaunt jetty]

What's your resultant equation

[formal terrace]

Wdym

Like what I’ve done so far

?

[gaunt jetty]

Sure

[grand gust]

yes, please show your work

[formal terrace]

Literally just a+b=10 😭

[lost sphinx]

then minimize a^2 + b^2

[grand gust]

so you know you should apply pythagoras. can you apply that first?