so we need dim(W) <= 3
#help-49
nimble kraken
midnight plankBOT
prime hornet
midnight plankBOT
Available help channel!
tidal turret
midnight plankBOT
tidal turret
elder knoll
(I'd recommend translating)
midnight plankBOT
midnight plankBOT
Available help channel!
twilit field
If $A$ is closed, then prove $A^C$ is open.
\
We will prove if $A^c$ is not open, then $A$ is not closed
\
\begin{proof}
Let $A^c$ not be open. Then, $\exists x \in A^c : \forall \varepsilon >0 V_{\varepsilon}(x) \nsubseteq A^c$. This implies for every point epsilon- neighbourhood, there exists a point in $(A^c)^{c}$, which is $A$.Using these points in $A$,. we can form a sequence, that converges to $x$, which is not in $A$.
\end{proof}
midnight plankBOT
If $A$ is closed, then prove $A^C$ is open.
\\
We will prove if $A^c$ is not ope...
grand pondBOT
wai
twilit field
we can form a sequence, that converges to x, which is not in A.
I'll have to prove this , right
but is the basic idea right
\begin{proof}
Let $A^c$ not be open. Then, $\exists x \in A^c : \forall \varepsilon >0 V_{\varepsilon}(x) \nsubseteq A^c$. This implies for every point epsilon- neighbourhood, there exists a point in $(A^c)^{c}$, which is $A$.Using these points in $A$,. we can form a sequence, that converges to $x$. Infact every seqeunce formed using these points converges ot $x$, as they all lie in $(V_{\varepsilon} (x) \setminus {x}) \cap A$. Thus $x$ is a limit point fo a seqeunce in $A$, but $x$ doesn't lie in $A$. Thus $A$ is not closed.
\end{proof}
grand pondBOT
wai
hard umbra
what is the sequence though
twilit field
what is the sequence though
Hmm?
hard umbra
you say "these points" as if that determines a sequence
twilit field
Right, I have to define the sequence properly
twilit field
so what is the sequence?
for each natural number we can define a_ eps to any term in the intersection of that eps neighbourhood and $A$
grand pondBOT
wai
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
twilit field
but not ssure of how to deal with eps <1
twilit field
epsilon
hard umbra
that's clearly not what i mean when i ask that question
you say eps, and yet you don't tell me what it is
is it a natural number?
twilit field
you asked me to defein the seqeunce
I'm saying for instance a_1 is any term in the intersection of the deleted 1-neighbouhhod of $x$ and A
grand pondBOT
wai
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
hard umbra
what is a_n? define it for each natural n
twilit field
what is a_n? define it for each natural n
$a_n$ is any term in the intersection of the deleted $n-$ neighbourhood of $x$ and $A$
grand pondBOT
wai
hard umbra
think about what you're trying to do, and think about whether that choice makes sense
what should the sequence converge to?
are your choices converging to what you want it to converge to?
twilit field
I think they do
hard umbra
A = (-inf, 0), A^c = [0, inf) is not open
take 0 to be the point in question
i will pick a_n = n/2
does that converge to 0?
is that in the n-neighbourhood of 0 for each n?
twilit field
no
hard umbra
I think they do
so try again
hard umbra
you want your sequence to converge to x
the neighbourhoods you pick had better get smaller not larger
twilit field
so I have to chose a countably infinite subset of $1$ for instance
grand pondBOT
wai
hard umbra
a subset of 1?
hard umbra
i mean
you can choose an infinite subset that falls within (1/2, 1]
that would be pointless
twilit field
Maybe each time we chose a 1/n neighbourhood
that should work by the archimedian property
hard umbra
yes do that
twilit field
yes do that
so I just define the seqeunce , right
hard umbra
well given the effort put in so far, i wouldn't put it as "just"
twilit field
We define $a_n$ to be any point in the intersection of the deleted $1/n$ neighbourhood of $x$ and A. As for every positive real number , there exists a smaller number $\frac{1}{n}$, less than it, we can find a candidate for any $\varepsilon$
grand pondBOT
wai
twilit field
I think this is misisng some formalism, not sure how to insert that
hard umbra
just pick a_n to be an element of (x - 1/n, x + 1/n) cap A, which you know to be non-empty by the assumption that A^c is not open
midnight plankBOT
midnight plankBOT
Available help channel!
quaint field
if I have two parallel lines and a point not on the lines, can I create a third parallel line going through that point with only a straightedge
midnight plankBOT
if I have two parallel lines and a point not on the lines, can I create a third ...
jaunty ivy
what do you mean by straight edge
jaunty ivy
so a single line going through the point that is also parallel to the other 2 parallel lines?
quaint field
yes
jaunty ivy
this is possible because you can choose any parallel line in the 2 dimensional plane. you can imagine a bunch of parallel lines grouped together so they fill the entire plane
since the point is on the plane (we call it the R^2 plane), there exists a parallel line that goes through the point
this is assuming our space is 2 dimensional
quaint field
this is possible because you can choose any parallel line in the 2 dimensional p...
i dont get it
quaint field
this is assuming our space is 2 dimensional
it is
jaunty ivy
look at these parallel lines
quaint field
yes
jaunty ivy
now imagine if i fill the whole plane with them
so i have every y intercept covered (the constant term)
quaint field
but how do i get those lines in the first place
because i only have two
so how do i make more
jaunty ivy
yes but we wanna see if there exists a parallel line to those lines that passes a point not on those 2 lines
quaint field
yes
bronze mason
I suspect they are asking how to construct this line using only a physical thing called a 'straightedge' (actual instrument)
quaint field
I suspect they are asking how to *construct* this line using only a physical thi...
yeah thats my question
quaint field
it can be any arbitrary point not on the lines
jaunty ivy
okay so we want another parallel line that satisfies this. parallel lines have the same gradient, we will use this
let the generic point be (a,b) not on the two parallel lines and let the gradient of the two parallel lines be m
quaint field
alright
jaunty ivy
we will have that the line equation of the other parallel line is
y=mx+c
where c is a value we need to find. hence we plug in the value
so we get b=ma+c
isolating c, we get b-ma = c
so our final equation is going to be
y=mx + b-ma
quaint field
i know how to get the equation of the line
but how do you actually make the line
jaunty ivy
you cannot if you dont have the parameters of m and the point (a,b)
quaint field
so it is impossible?
jaunty ivy
unless you wanna draw infinite lines
quaint field
oh
livid python
<@&268886789983436800>
livid python
hell yeah I won
jaunty ivy
we all won!
livid python
hell yeah
onyx slate
Stolen valor ahh
quaint field
if I have two parallel lines and a point not on the lines, can I create a third ...
so the answer to this is no?
olive matrix
you so 2 thousand and late
jaunty ivy
so the answer to this is no?
if you have the parameters then you can, ie if you know the gradient of the parallel lines and the point you want to go through
jaunty ivy
no problem!
quaint field
.close
midnight plankBOT
sharp coral
are you allowed the use of a compass?
livid python
i don't think so given that they said "with only a straightedge"
jaunty ivy
i mean even with a compass, the point they want to pass through is not even known?
sharp coral
there are several known constructions of a line parallel to a given line passing though a point using compass and straightedge
but i'm not aware of any using only a straightedge
actually here is a relevant SE post:
https://math.stackexchange.com/questions/3495183/draw-a-parallel-line-with-only-a-straightedge
midnight plankBOT
Available help channel!
mint ravine
Can someone please help me on part B
midnight plankBOT
Can someone please help me on part B
mint ravine
I got the answer 512/729 but I don't know if it's right
Here's my thought process:
First, we have to calculate the probability of getting all heads on the first turn (as indicated by part a) which is (2/3)^3=8/27
Then, to calculate for part B, you have to add 8/27 and the probability of getting heads on her second turn
and that's how i got 512/729
exotic pendant
i think how it works is you get the probability of it happening in the first rurn and then add the probablility of it happening on the second turn if the first turn fails
mint ravine
yeah i know
i also did cases for that
here are my calculations, feel free to ask if you are confused (ik it's a little messy)
exotic pendant
i'm kinda confused but that's more just because i use a different method for this 😅
grand pondBOT
migu
exotic pendant
then subtracting that from 1 (to get the chance of in not happening) you do $1-\frac{8}{27}=\frac{19}{27}$
grand pondBOT
migu
exotic pendant
now, you multiply this chance by the chance of flipping all heads to see the chance of it happening on turn 2 and not turn 1
,w 19/27*8/27
grand pondBOT
exotic pendant
actually now i'm doubting myself sorry D:
let me just double check
okay yeah
so now you add that to the original getting
,w 8/27+152/729
grand pondBOT
exotic pendant
and i believe that's it
mint ravine
and i believe that's it
but can't you have multiple instances of the same case
for instance, T-T-H could be ordered in 3!/2 or 3 ways
T= Tails, H=Heads
exotic pendant
uhhhh let me just
i'm pretty sure the chance of H-H-H is just (2/3)³
or 8/27
because it's a ⅔ for thrice
mint ravine
yeah it is
but e.g. you get T-T-T
it's also 8/27 chance you to get 3 heads after rolling them
but to get 3 Tails in the first place is (1/3)^3
so then i think you multiply 8/27*1/27
to get the probability
exotic pendant
if you want to do it the tedius way you could just do that for all possible permutations
mint ravine
tbh idk im not the best at probability
exotic pendant
but i just did it for any case that isn't H-H-H on turn 1
me neither :p
hence all the second guessing in the middle of explaining x3
midnight plankBOT
@mint ravine Has your question been resolved?
celest pebble
Can someone help me understand Stokes Theorem. To my understanding, stokes theorem is making the closed contour of a surface into a double integral. Im confused about the unit normal vector and how it changes depending on the orientaion.
midnight plankBOT
Can someone help me understand Stokes Theorem. To my understanding, stokes theor...
pure wraith
Do you have an example?
celest pebble
ya its from james stewarts multi calc book, question 9
in the answer sheet, im confused why there are so many negative signs, I understand how the curl is a negative but . . .
celest pebble
???
pure wraith
what exactly is your question lol
is it just "why are there so many negative signs?"
celest pebble
I read that i have to dot the curl of F with the unit normal vector
im not sure if im doing it wrong or what, but im getting a different equation from the answer sheet
so what exatly is the next step after i find the curl and bounds?
fallow scarab
im not sure if im doing it wrong or what, but im getting a different equation fr...
!show
midnight plankBOT
im not sure if im doing it wrong or what, but im getting a different equation fr...
Show your work, and if possible, explain where you are stuck.
@celest pebble Has your question been resolved?
undone heath
Is my solution for this integral correct
midnight plankBOT
Is my solution for this integral correct
pure wraith
Is that $\int \frac{\dd{x}}{x^3\sqrt{x^2 - 9}}$?
grand pondBOT
jewels!
undone heath
And restrict it to real
Bcs it’s simple
I also use trigonometric functions or simple functions at large
By simple function I mean any linear combination of indicators
pure wraith
seems like you've gone in circles
undone heath
why are you so obsessed with hyperbolic subs though
Another reason is that hyperbolic substitution gives exponential function allowing rapid manipulation
pure wraith
also what happened to the x^3
undone heath
Yes I forgot it
I will do it again, I missed it
A minute
And you’ll know one day how nice it is to have e^x in an integral
pure wraith
Maybe x = 3sec(u) is a better sub here though
grand pondBOT
jewels!
midnight plankBOT
@undone heath Has your question been resolved?
undone heath
Maybe x = 3sec(u) is a better sub here though
You’re right
I am too dumb and eitherway I haven’t been very good with this 😭😭
And I am actually a noob in math in fact
Another reason for hyperbolic substitution is that I can always easily expend it by Taylor so at worst I got numerical result
This is possible since continuous function is Borel measurable
So it becomes polynomial integral
undone heath
Maybe x = 3sec(u) is a better sub here though
Your method is brilliant I solved it with it with ease
Im happy now
.close
last slate
since 1+4 = 5, i know that if num in form ABC14, then A+B+C = 3x, 3x+1 for cases that dont work
A+B+C = 3x-2 for cases that do work
do i literally have to count
midnight plankBOT
since 1+4 = 5, i know that if num in form ABC14, then A+B+C = 3x, 3x+1 for cases...
floral apex
hmm you could use PIE
but that also involves counting
I don't think counting is that hard here
mortal mirage
a+b+c = 1 mod 3
last slate
a+b+c = 1 mod 3
not all the time i dont think
floral apex
,w 99999/3
grand pondBOT
last slate
😭
mortal mirage
not all the time i dont think
how come?
grand pondBOT
Result:
45
its not really that many cases
floral apex
,calc 45/3
grand pondBOT
Result:
15
?
last slate
15 cases in what context wait
last slate
maybe im dumb though <:blobsweat:584506239661113371>
no no im just not following
floral apex
sum of digits divisible by 3 -> divisible by 3
the lowest it could be is 0 (or 3)
the highest it could be is 45
last slate
the sum of A+B+C can be 1,4,7,10,13,16,19,22,25
floral apex
45/3 = 15 so 15 cases of sums of digits
last slate
45/3 = 15 so 15 cases of sums of digits
wait whered' you get 15
last slate
idk i didnt know where to go from there icl
mortal mirage
but you need to account for repeating numbers no
floral apex
repeating?
mortal mirage
like 7=2+2+3
floral apex
you dont need to worry about that
theres a pattern in the distribution of the weights
floral apex
you need to count the ways you can distribute each of those masses across 3 bins
mortal mirage
oh wait stars and bars whoops
floral apex
its a little confusing since you have to move from numbers to like
unmarked balls into distinguishable urns
basically
last slate
oh uh
mortal mirage
mhm
last slate
and then add those tg
floral apex
yea
last slate
ouhh
floral apex
its not the fastest way probably
but also its not that hard of a problem
so why not
last slate
yeah i was wondering if there was a faster way
floral apex
i wonder if PIE is faster 
last slate
HM
floral apex
i think it is
well, kind of
you end up with the same stars and bars problem
but theres more cases
floral apex
maybe i can scribble lemme see
last slate
lowk crashing out cuz this was supposed to be the easy unit no one struggled with and here i am 😭
mortal mirage
wait whats PIE (im sorry idk anything)
last slate
100x + 14 = 0 (mod 3)
floral apex
im just being dumb
last slate
wait whats PIE (im sorry idk anything)
princ incl excl
floral apex
yea
last slate
100x = 1 (mod3)
floral apex
but it doesnt make any sense here
mortal mirage
princ incl excl
ohh thanks
floral apex
its easier to use it when you want one of the outside sets
last slate
uhhh
last slate
ohh thanks
np
mortal mirage
ik the theorem but idk the abbrieviation lol
last slate
its easier to use it when you want one of the outside sets
yeah idk how id use pie
um
100x = 1 (mod3)
mortal mirage
given that tho i think stars and bars is the fastest way
last slate
and x = 1 (mod 3)
floral apex
yea it works out fine
midnight plankBOT
midnight plankBOT
Available help channel!
rugged pumice
what do you call this? factoring? but it's not really factoring?
$\ (a-b)\to (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})$
midnight plankBOT
what do you call this? factoring? but it's not really factoring?
$\\ (a-b)\to (\...
rugged pumice
expanding?
grand pondBOT
someone1010
buoyant linden
what do you call this? factoring? but it's not really factoring?
$\\ (a-b)\to (\...
difference of two squares
i suppose it's still a form of factoring tho
it wouldn't be wrong to say it is factoring
jaunty ivy
for a,b positive
√(a)^2 = a
and √(b)^2 = b
hence you get the difference of squares
fathom onyx
yes*
rugged pumice
like if im using this in a proof i just say by difference of 2 squares?
sounds kinda weird to me idk
fathom onyx
You can just state it
jaunty ivy
for a,b positive
√(a)^2 = a
and √(b)^2 = b
hence you get the difference of squar...
you can say this
fathom onyx
The RHS expands and simplifies to the LHS, so there's nothing against it
rugged pumice
you can say this
this seems kinda excessive
idk im just gonna say expands, thanks yall
.close
midnight plankBOT
fathom onyx
idk im just gonna say expands, thanks yall
"expands" is the wrong way 'round
rugged pumice
huh?
fathom onyx
"expands" is what we say when multiplying out brackets
You're literally doing the opposite of this
Since the LHS is algebraically the same as the RHS, you can just state the equivalence without anything else
rugged pumice
oh true
fathom onyx
$a - b = (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})$
grand pondBOT
Waes (Wires)
fathom onyx
Nothing further needs to be said lol
midnight plankBOT
Available help channel!
spiral talon
How can I use the identity theorem to show that a holomorphic function with $$ h(\frac{1}{2n}) = \frac{1}{n} = h(\frac{1}{2n - 1 }) $$ exists/does not exist?
I understand the meaning of the identity theorem -- if two holomorphic functions coincide at infinitely many points in a small enough region on the domain they're identically equal on the whole domain -- but I dont know how to work with it.
midnight plankBOT
How can I use the identity theorem to show that a holomorphic function with $$ h...
grand pondBOT
Edlingem
pine wave
How can I use the identity theorem to show that a holomorphic function with $$ h...
start with a guess, do you reckon it exists or not
spiral talon
I think it doesn't
pine wave
okay, so maybe you can get a contradiction somewhere
if you just work with h(1/(2n))=1/n maybe you get some function
midnight plankBOT
@spiral talon Has your question been resolved?
spiral talon
i dont get it?
solid iris
i dont get it?
pick an easy function g st g(1/2n)=1/n
spiral talon
g(z) = 2z?
solid iris
g(z) = 2z?
now what can we say about g and h?
spiral talon
They coincide at every 1/2n so there is an accumulation point at 0
So h and g must be identically equal on C
solid iris
yes can u see how to finish?
spiral talon
Ah and for the other sequence i can choose g2 s.t. g2(1/(2n-1)) = 1/n
solid iris
no need
spiral talon
h(1/2) = 2(1/3) but also 1/2 by the second condition
solid iris
ya
spiral talon
Nice, thank you very much!
solid iris
np 🙂
spiral talon
I would like to keep the channel open for a moment bc i got another similar problem. But I will try this alone first… just if any more questions occurr on the way
solid iris
h(1/2) = 2(1/3) but also 1/2 by the second condition
wait how did u get this
u mean h(1/3)?
spiral talon
pardon, yes
solid iris
all good then
spiral talon
I got the second problem, it was really straight forward. Thanks again
solid iris
np 
spiral talon
.close
midnight plankBOT
midnight plankBOT
Available help channel!
tired maple
the og one got closed even tho the guy was done with his question
midnight plankBOT
the og one got closed even tho the guy was done with his question
lyric charm
this looks correct to me
steel marten
apply d less than zero
tired maple
the other guy said i solved it wrong
lyric charm
i verified it through desmos
lyric charm
it really is just k > 1/3
lyric charm
maybe lordfelix messed it up himself
tired maple
and for 11
i got to 25+8k^2
and my teacher told me to complete square
but that guy told me its 35+8k^2>0
steel marten
delicate sage
!occupied
midnight plankBOT
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).
delicate sage
i got to 25+8k^2
You did it correct
delicate sage
i got to 25+8k^2
25+8k²> 0, solve this
tired maple
everything except for the last line has been done by me
delicate sage
i got square root of -25/8
Think, logically
delicate sage
Can 8k² be lesser than 0 ? For real k ?
delicate sage
that seems to be the only logical option to do
Ofc not.
delicate sage
Can 8k² be lesser than 0 ? For real k ?
Least value of 8k² is 0, and 25>0, so isn't the eqn true for all k
tired maple
Can 8k² be lesser than 0 ? For real k ?
no because anything squared will be >0
thank you
.close
midnight plankBOT
midnight plankBOT
Available help channel!
rare maple
midnight plankBOT
cedar coral
the other guy said i solved it wrong
.yeah my bad, i thought it was the same conditions as before with 2 real. Sorry.
midnight plankBOT
desert siren
It's 11 blud
lyric charm
It's 11 blud
timezones?
desert siren
he's doing jee
steel marten
no because anything squared will be >0
tired maple
.yeah my bad, i thought it was the same conditions as before with 2 real. Sorry.
okay
sorry for the misunderstanding
steel marten
misunderstanding
tired maple
?
steel marten
i ask you what misunderstanding
midnight plankBOT
Available help channel!
magic remnant
midnight plankBOT
magic remnant
chain rule gets f'(x)/2(sqrt(f(x)))?
desert siren
Indeed
magic remnant
i don't understand what do do after that
desert siren
Well at x=3, can you see that the blue line is a tangent to the red curve?
magic remnant
yeah
desert siren
So f'(3)= slope of the blue line
magic remnant
-2/3
magic remnant
no?
desert siren
Why?
magic remnant
f(x) is still unkown
desert siren
Well, can't you find f(3) by looking at the graph?
desert siren
Yep
magic remnant
is that the answer?
magic remnant
alright
i had more difficulty in understanding the question than the algebra
ugh
thank you
desert siren
yw 
magic remnant
.close
desert siren
!done
midnight plankBOT
If you are done with this channel, please mark your problem as solved by typing .close
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mint ravine
Can someone please help me on this question
midnight plankBOT
Can someone please help me on this question
mint ravine
I initially got the answer 3/8 but it was wrong
Can someone please help
I tried breaking it into two cases for each possible sum
10+10+10
10+15+5
<@&286206848099549185>
mortal mirage
have you done casework?
mint ravine
yeah i said it in the explanation
mint ravine
4 cases
mortal mirage
i think youre missing half of them
mortal mirage
does the order in which she visits the square matter
grand yarrow
Can someone please help me on this question
mint ravine
does the order in which she visits the square matter
oh wait it does yes i think
grand yarrow
Can someone please help me on this question
but it is in Romanian
mint ravine
bro open your own channel
mortal mirage
Can someone please help me on this question
!occupied
midnight plankBOT
Can someone please help me on this question
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).
grand yarrow
bruh
mortal mirage
oh wait it does yes i think
So she has 4 choices for her initial pick and 2 for her second pick, and the third she only has one choice
which makes 8
mint ravine
who said she couldn't go back?
mint ravine
i think it's 4^2=16
mint ravine
yeah i think it's 4 too
mint ravine
yes
mortal mirage
👍
midnight plankBOT
sharp coral
<@&268886789983436800>
midnight plankBOT
livid python
<@&268886789983436800>
cloud for the rescue 🔥
midnight plankBOT
Available help channel!
steel rivet
let a,b,c are real numbers such that 4a+2b+c = 0 and ab > 0. Then the equation ax2 + bx + c =0 has
a. real roots b. imaginary root c.exaclty one root d. infinite roots
midnight plankBOT
let a,b,c are real numbers such that 4a+2b+c = 0 and ab > 0. Then the equation a...
burnt wing
let a,b,c are real numbers such that 4a+2b+c = 0 and ab > 0. Then the equation a...
You want the answer or..?
civic lynx
-# no solutions pls just try to walk through stuff
burnt wing
Have you attempted it
steel rivet
yes
burnt wing
What did you get to?
burnt wing
Huh
steel rivet
i tried manipulating the given eqn
burnt wing
Can I see your working out and then I can see where u went wrong
steel rivet
give me idea behind it
woeful turret
let a,b,c are real numbers such that 4a+2b+c = 0 and ab > 0. Then the equation a...
i would take a=-1,b=-1
so c=6
steel rivet
i would take a=-1,b=-1
what is reason for it
burnt wing
Ab is greater than zero so sign is the same
woeful turret
what is reason for it
no reason, condition should be true for all values of a and b so pick some values which satisfy condition
steel rivet
ok
civic lynx
well, the motivation is based around the discriminant, i.e. $\Delta = b^2 - 4ac$
do you know of the relationship between discriminant and roots?
burnt wing
the discriminant isn’t the square root of b^2 -4ac it’s just b^2-4ac
civic lynx
oh true ty
grand pondBOT
haseeb
steel rivet
yes
steel rivet
do you know of the relationship between discriminant and roots?
yes
if we substitute x = 1 and x =2 it gives zero
even for x = 1,2,3,4....
civic lynx
great! so you could manipulate the discriminant by subbing in the 4a + 2b + c = 0 or the ab > 0, and if it ends up postive, zero, or negative, you know which one it is
steel rivet
i think the same but didnt try to do
civic lynx
i think the same but didnt try to do
fair enough lol, it is a little long for a multiple choice question
burnt wing
Yeah fr what exam is this
steel rivet
just exercise question
steel rivet
Yeah fr what exam is this
mock tests in classroom
burnt wing
Uk or USA
steel rivet
India
burnt wing
Oh alr
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stable quartz
Need help finding answer 🥲🙏
midnight plankBOT
Need help finding answer 🥲🙏
gaunt nimbus
What’s the amplitude
What’s the period
Is it a sin or cosine
(Evidently there’s no phase shift and it is centered about y=0)
stable quartz
Amplitude is 3 😎
gaunt nimbus
How about period
gaunt nimbus
(Evidently there’s no phase shift and it is centered about y=0)
(But there is reflection👀..)
stable quartz
OHHHH
gaunt nimbus
Idk g
How long does it take to complete one cycle
Compare it to how long it usually takes
stable quartz
It takes 2 on x-axis
stable quartz
2?
gaunt nimbus
2 = 2pi/B
stable quartz
Ahh, so B= 2pi/2
stable quartz
Oh
stable quartz
Yeah
gaunt nimbus
About which axis?
stable quartz
I didn’t notice the reflection, I just said yeah, where is the reflection?
stable quartz
Sin
gaunt nimbus
U sure?
stable quartz
Nah, I was just testing you, it’s a cosine
stable quartz
-3
gaunt nimbus
What’s sin(0)
stable quartz
0
gaunt nimbus
From this graph is it 0?
stable quartz
Nope 🙂↔️
stable quartz
Ohhhh
gaunt nimbus
Generally cos(0) =1
So
3cos(pix)) where x=0 is 3(1) =3
What reflection would it be
So that the graph outputs -3 at the start
gaunt nimbus
What reflection would it be
About which axis (x or y)?
stable quartz
Uhhh, y
gaunt nimbus
Why did u think that?
stable quartz
Uhh, I guessed
gaunt nimbus
Ye
stable quartz
AYYEEEE
gaunt nimbus
But it’s not the correct reflection
stable quartz
Dang
gaunt nimbus
Nor is it the correct reason but it’s correct that y changes but x remains the same
stable quartz
WHA, then why’d you say ye 🥲
gaunt nimbus
I mean it’s a correct statement but for a different question
😔
Anyway
It’s a reflection about x axis
stable quartz
Ohhh
stable quartz
Ima try
So far I’ve got 3cos(pi(x))
Therefore
It’s -3cos(pix)
I got it right
Thank you so much Albert Einstein
Have a good one
You smart asf
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raven zenith
WAIT I NEED UR HELP WITH ANOTHER Q
You can post it, another helper can help you
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molten bay
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molten bay
113
night hawk
If it always has the same sign, it’s either strictly positive or strictly negative
Neaning it as no roots
Gtg
hard shard
use the discriminant
signal holly
If c is +ve, then, a>0 and D<0. If c is -ve, then a<0 and D<0. So, D<0 common, b^2<4ac
molten bay
Discriminant is b^2-4ac
molten bay
If c is +ve, then, a>0 and D<0. If c is -ve, then a<0 and D<0. So, D<0 common, b...
How can you conclude that?
If Y>0 then surely it will not have real solutions so D<0
If Y<0 same thing
So D<0 always
b^2-4ac<0
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fringe pine
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fringe pine
the problem is that idk how to get it “in terms of x”
i doubt anyone’s on during the summer though 
latent wadi
oh my god
lethal path
instead of using trig, try using coord geo
you can let x = 1 because of area similarity
fringe pine
you can let x = 1 because of area similarity
lethal path
then you know the coordinates of points P and C, if you set point D to be the origin
latent wadi
tbh u could use trig, but all u need is knowing cos and sin of 0, 30, 45, 60 ,90
fringe pine
that makes sense
lethal path
I mean, yes you could use trig, but
you need the coordinates anyways I'm pretty sure
lethal path
<a:shocker:1373657049468108861> <:shock:1107717300045496400>
yes then multiply by x^2 at the end
fringe pine
i think i’ll only use geometry
my friend said no trig, calc, or anything other than geo
that was the deal
(don’t tell her i’m getting outside help) 
wait how am i supposed to solve this with no trig?
pine wave
Connect C to midpoint AB then whack for length ratios and use similar triangles?
fringe pine
Connect C to midpoint AB then whack for length ratios and use similar triangles?
i forgot midpoint formula
oh wait
i don’t need that
fringe pine
Connect C to midpoint AB then whack for length ratios and use similar triangles?
wait now i’m lost
WAIT A MINUTE
I HAVE AN IDEA
nvm won’t work
fringe pine
yes then multiply by x^2 at the end
i have a feeling it was just x^2, but why multiply it at the end?
lethal path
i have a feeling it was just x^2, but why multiply it at the end?
so you square the length ratio to get the area ratio
easiest to see with a square, so if two squares have side lengths 4 cm and 1 cm, their areas are 16 cm^2 and 1 cm^2
but then for other shapes, like a triangle where A = 1/2 bh, the 1/2 will cancel out when you divide the two sides, so the shape doesn't matter
so if the length ratio is x/1 = x
the area ratio must be x^2, or x^2 times larger
lethal path
that's insane
fringe pine
yes then multiply by x^2 at the end
i restarted and back tracked. i set x = 1 and labeled points P, C, and D with D at the origin
now we need to find the area, but in terms of x
lethal path
there must be a clever way to show that the area is 6 * x^2 systematically but you already have the answer
wait
I know it's 3, 4, 5 cause of trig ratios, but it looks like you just guessed
no hold on
fringe pine
I know it's 3, 4, 5 cause of trig ratios, but it looks like you just guessed
i didn’t even make the connection that 345 has an area of 6 to begin with 
i just luckily guessed
unluckily now
lethal path
wait
fringe pine
if x = 1, that means our area isn’t 6
but with the points we have, we can use distance formula to find BP
the only problem then is BQ
i’m going to bed my other friend solved it before i did
i was very close though
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twin star
Fourier Transforms, Circulant matrices
A NxN matrix is circulant if you can get the next row by shifting the current row by 1.
We can define a circulant matrix with only its first row, and they have the nice property that if you take the fourier transform of the first row and the input, multiply them elementwise, and do an inverse fourier transform, you get the vector-matrix product of the input with the whole circulant matrix.
I've got it working just fine in the regular circulant matrix case. The linked file contains my code and runs as is (I got other tests not shown here to check for correct output). Take a look at it, I explain something about reduction in frequency domain in there (the einsum in the circulant_mul function) that I refer to later.
If I talk about a (b, i, o) shaped object, it means it's a tensor of 3 axes with the axes length being (batch_size, input_dim, output_dim). In this context, i = o (circulant matrices are square). The FFT on a vector of N reals has size (N/2+1).
I have 4 "scenarios" (the test functions at the end):
- regular circulant matrix: kernel is (1, i), reduction over the 1 axis (so no reduction lol)
- multiple circulant matrices: kernel is (m, i), reduction over the m axis
- multiple circulant matrices: kernel in frequency domain is (m/2+1, i/2+1), reduction over the m/2+1 axis
- multiple circulant matrices: kernel in frequency domain is (o/2+1, i/2+1), reduction over the i/2+1 axis
All scenarios give us a (b, i) output. First two are easy (circulant matrix-vector product, and sum of circulant matrix-vector products).
I don't really understand what the other two do. Third one is similar to second one, except the kernel in frequency space is half the size. Fourth one reduces over another axis (btw we can use this to have the output be (b, o), with o!=i). What do they do exactly? What does it mean to have half the kernel size in frequency space? What does it mean to reduce over the other axis?
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# Fourier Transforms, Circulant matrices
A NxN matrix is circulant if you can g...
@twin star Has your question been resolved?
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@twin star Has your question been resolved?
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@twin star Has your question been resolved?
desert siren
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@twin star Has your question been resolved?
twin star
Try <#576511214025310260>
I cross linked there, but no one answered 🤷
Should I copy-paste the whole thing?
fallow scarab
Nah link is fine
Just wait or repost on stack exchange with the right tags
Random matrix theory maybe
twin star
Yeah I was thinking that for this kind of question discord isn't the best place
fallow scarab
Correct it's not
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molten bay
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https://discord.com/channels/268882317391429632/1018700692027883620/139029794467...
twilit field
I would like to prove if $K$ is closed and bounded then every open cover for $K$ has a finite subcover
midnight plankBOT
I would like to prove if $K$ is closed and bounded then every open cover for $K$...
grand pondBOT
wai
twilit field
!status
midnight plankBOT
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
1
tidal spade
K is a subset of R^n, I suppose?
twilit field
R
tidal spade
what tools do you have available? Bolzano-Weierstrass?
twilit field
Sure
next bramble
I would like to prove if $K$ is closed and bounded then every open cover for $K$...
Does it need to be R, or just a connected space?
sage helm
i did it by proving it for a closed interval first
twilit field
Does it need to be R, or just a connected space?
R
twilit field
i did it by proving it for a closed interval first
hmm
sage helm
and then you choose that closed interval such that K is a subset of that interval and you can do some work and extend it
doesn't that exercise give you steps
twilit field
I didn't see that exercise 😭
twilit field
I'm in the theory
tidal spade
of bolzano weistres?
What's the statement of Bolzano-Weierstrass that you know of?
twilit field
What's the statement of Bolzano-Weierstrass that you know of?
Eveery convergent seqeunce has a monotone subseqeunce
wait, that's it, right, or am I mxing theorms up
every bounded seqeunce has a convergenet subseqeunce
that's better
So I prove that every bounded set has a monotone subseqeunce contained in it?
tidal spade
that's better
Do you also know other characterizations of "every open cover has a finite subcover"?
twilit field
Do you also know other characterizations of "every open cover has a finite subco...
K is compact, but haven;t proven that
tidal spade
K is compact, but haven;t proven that
What about sequential compactness? (K is compact iff every sequence in K has a convergent subsequence in K)
tidal spade
What about sequential compactness? (K is compact iff every sequence in K has a c...
(which some people would also call Bolzano-Weierstrass)
twilit field
What about sequential compactness? (K is compact iff every sequence in K has a c...
I have
tidal spade
I would like to prove if $K$ is closed and bounded then every open cover for $K$...
Now, assume that K is closed and bounded. Try to show that every sequence in K has a convergent subsequence in K. This is equivalent to the fact that K is compact (every open cover has a finite subcover).
twilit field
Now, assume that K is closed and bounded. Try to show that every sequence in K h...
hmm
the thing is
I don't know that K is compact implies that every open cover has a finite subcover
tidal spade
that's one definition of compactness
tidal spade
What about sequential compactness? (K is compact iff every sequence in K has a c...
It's not much of a proof if you can use this tbh
(that's why I said direct corollary)
You just have to recall another characterization of closedness
twilit field
I don't follow
tidal spade
I don't follow
Start with the fact that K is bounded. What can you say about sequences in K?
twilit field
they are bounded
tidal spade
So?
twilit field
There exists a monotne subseqeunce in it taht converges to K?
runic hamlet
wai is this an exercise in your book or a theorem
twilit field
A theorm, but this part of the proof is an exercise, abbott has given a hint in the exercise, do I read thgat?
runic hamlet
jeez
yes of course
hints are the authors way of telling you: unless you are very good, you will not get this on your own
(there are of course exceptions, but hints are there for a reason)
grand pondBOT
twilit field
,rotate
grand pondBOT
twilit field
,rotate
grand pondBOT
runic hamlet
(you can click on the reacts to rotate it further)
twilit field
so now I do these steps
runic hamlet
yes
twilit field
okay, I'll do them and then opena a channel then?
runic hamlet
sure however you want
bear in mind, it is normal and expected that exercises can take hours
so take your time
before you come back
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tardy bloom
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hard burrow
Can you tell me where the e^{3x} went in the denominator
tardy bloom
It's 1, isn't it?
hard burrow
You partially evaluated the limit?
tardy bloom
Yeah
lim x→a( fg)(x) = lim x→a f(x) × lim x→a g(x), right??
Pls ping when replying
hard burrow
lim x→a( fg)(x) = lim x→a f(x) × lim x→a g(x), right??
assuming both limits exist yea
hard burrow
Does this work?
can you clarify what the denominator after the e^{3x} disappeared
is that supposed to be sin x
tardy bloom
No just x, sinx= x for x tending to 0
lim x→0 sinx/x =1
I just basically multiplied and divided the denominator by x and again partially evaluated the limit
tardy bloom
lim x-→0 e^-1/x2 / x =0
I evaluated it somewhere else, put x=1/u then lhopital
hard burrow
edit: i didnt see the negative sign
hard burrow
what happened to the initial conditions?
tardy bloom
Yeah I didn't that's why I don't know whether this is correct or not
hard burrow
Does this work?
this part seems fine to me
hard burrow
Does this work?
no this is incorrect actually
you forgot negative sign in your integrating factors
tardy bloom
?
hard burrow
y_1 = e ^ integral -(sin^2x)
tardy bloom
I integrated it, didn't use integrating factor
civic lynx
aren't these DEs seperable, i.e. $\frac{y_1'}{y_1} = \sin(x)$, moved over to the other side?
grand pondBOT
haseeb
civic lynx
so then you'd get $y_1 = e^{\int sin(x) \mathrm{d} x}$
grand pondBOT
haseeb
civic lynx
sry my brain defaults to linear
all g, lowkey linear >>>
solution looks good to me! you could solve for ICs like i said, but i think you can just say e^C gets absorbed into the vanishing limit
hard burrow
if i were you id just solve for the the explicit forms using the IC
but yea youre right
they kinda dont matter xd
civic lynx
its good DE practice :)
tardy bloom
that seems too lengthy for me, I am looking for short solutions
civic lynx
the one you wrote out? id argue its pretty short: 1 line to solve 3 DEs and 1 line to evaluate a limit
tardy bloom
I only have like 4-5 mins for 1 question ☠️
civic lynx
i would give this 7-10 minutes as a teacher, owing to the 2 points
oh and i personally would mention that e^{-1/x^2} > x so it will tend to 0 quicker
hard burrow
I only have like 4-5 mins for 1 question ☠️
i think what you did is doable in 5 min
midnight plankBOT
tardy bloom
oh and i personally would mention that e^{-1/x^2} > x so it will tend to 0 quick...
Wait, how can you say that?
It was pretty tough to solve that limit ngl
hard burrow
Wait, how can you say that?
because exponentials grows faster than polynomial
ln << x^n << n^x << x! << x^x (as x -> inf) (other way for x->0)
civic lynx
Wait, how can you say that?
aa sorry, e^{-1/x^2} < x near 0 D:
wheat cipher
who needs a tutor?
tardy bloom
ln << x^n << n^x << x! << x^x (as x -> inf) (other way for x->0)
Is this generalised? Like can I use this to solve limits??
wheat cipher
Is this generalised? Like can I use this to solve limits??
whats the problem?
tardy bloom
This
civic lynx
Is this generalised? Like can I use this to solve limits??
ish, it can still fail, especially around 0
but for infinity, yeah
tardy bloom
So best to avoid it?
civic lynx
i think it's a good starting point, especially in a test environment
tardy bloom
Okay
hard burrow
Is this generalised? Like can I use this to solve limits??
as x goes to infinity, yes
lim x->inf of lnx/x = 0
its very cool
you shouldve seen this in calc 1
tardy bloom
you shouldve seen this in calc 1
I'm not from the usa, that system doesn't work here 😕
hard burrow
I'm not from the usa, that system doesn't work here 😕
me neither
i mean, hey now you know
tardy bloom
me neither
I'm not in college either lol, that's why I Don't know that ig