@tidal viper compare this and what you sent earlier
#help-27
stark parcel
wraith wing
stark parcel
that's an assumption you be making~
rmb for such questions, there is a universal set U which most of the time contains those randos that do not meet any condition!
:)
tidal viper
oh i thought of that but didn't really go that way
stark parcel
@tidal viper so solved?
once u get the correct G, the 2nd part is way easier
actually i dont get why they had to say "exactly 1 mode" is 90, i dont think this info will be used anywhr lol
unless it's meant as a distraction
devout snowBOT
@tidal viper Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
quasi jacinth
can someone explain whats done here ?
devout snowBOT
can someone explain whats done here ?
quasi jacinth
pseudo basin
looks like we're calculating the image of this linear map L
did you want an explanation of the algebra in the middle line specifically?
quasi jacinth
did you want an explanation of the algebra in the middle line specifically?
yea
how did we get x(1,1)
stone stump
can you not see why that step is valid?
or can you not see why we have done that step?
quasi jacinth
can you not see why that step is valid?
this
pseudo basin
presumably the goal's to find a basis for Im(L)
quasi jacinth
ok so the third = how is it done
stone stump
can you go from right to left?
quasi jacinth
can you go from right to left?
yes
stone stump
whats that r thing
its introducing a new variable
stone stump
y-3/2z is some real number
we are calling that real number r
we dont actually get more options by having two variables
so lets get rid of them and have only one variable
one variable is easier to deal with than two
quasi jacinth
and now how do we conclude its R^2?
stone stump
(1,1) and (0,-2) are spanning R^2
you can write every vector (a,b) as some linear combination of them
quasi jacinth
yea because they are linearly independent right?
stone stump
yes
devout snowBOT
supple knot
<@&268886789983436800> spam
devout snowBOT
Available help channel!
idle dune
hey
idle dune
29:18
feral flame
whats the ques sir
feral flame
im sorry cant help
lament prawn
set each variable to 0 and draw the plane of the remaining variables
so you will have 3 different graphs
lament prawn
about what
lament prawn
go on desmos and try graphing x = y^2
idle dune
ok i did
lament prawn
yeah do you see how it's similar to x = z^2
idle dune
no
lament prawn
z is your vertical axis
idle dune
lament prawn
in the x-y plane, y is your vertical axis
devout snowBOT
half quarry
E(t) = T × C × R
Need help with this in how it can function in closed systems
It works in open, very well, but closed yields interesting information.
Ohh I figured out this
My bad didn’t meant to do this in your stuff
vestal oyster
E(t) = T × C × R
Need help with this in how it can function in closed systems
...
what is this bro
half quarry
Emergence formula
I’m a psy major who accidentally discovered its methods
Need validation I’m not going crazy 🌀
vestal oyster
but you didn't specify what E, T, C and R mean
devout snowBOT
Available help channel!
vestal oyster
and I am not psy major
devout snowBOT
and I am not psy major
wind mason
blud
"After further review, the ruling on the field stands: I shall close this help channel"
.close
ocean scroll
In attempting to solve #help-44｜stanley-🌲-v2-dans message, I created some data like this:[[[2, 3, 4], [2, 3, 4], 1], [[2, 3, 4], [2, 3, 4], 1], [2, 3, 4]] [[[2, 3, 4], [2, 3, 4], 1], [[2, 3, 4], [2, 3, 4], 1], [2, 3, 4]] [[[2, 3, 4], [2, 3, 4], 1], [[2, 3, 4], [2, 3, 4], 1], [2, 3, 4]] [[2, 3, 4], [2, 3, 4], 1]On this data, I do an algorithm:
- Choose any triplet of duplicate rows (Each row will be of the form [a,b,c])
- Replace each rows with one of the a, b and c (We can choose which row gets which part, giving 6 choices at each step)
- Repeat until every row is a number
[[[2, 3, 4], [2, 3, 4], 1], [[2, 3, 4], [2, 3, 4], 1], [2, 3, 4]]
[[[2, 3, 4], [2, 3, 4], 1], [[2, 3, 4], [2, 3, 4], 1], [2, 3, 4]]
[[2, 3, 4], [2, 3, 4], 1]
Choose any order for the unpacking
[[2, 3, 4], [2, 3, 4], 1]
[[2, 3, 4], [2, 3, 4], 1]
[2, 3, 4]
[[2, 3, 4], [2, 3, 4], 1]
Choose any order for the unpacking
[2, 3, 4]
1
[2, 3, 4]
[2, 3, 4]
Choose any order for the unpacking
2
1
4
3```In this example, I have n=4 numbers, and I believe the depth of the data is O(n).
What is the fastest approach to decide if all permutations of n numbers (S_n) is inside my data? Can it be done in polynomial time, or can it be shown to be exponential (or worse?!)?
There is also a variant of the problem where you are given multiple of these and you have to check if the union is S_n (Which is the original problem)
devout snowBOT
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).
bleak stag
Mb
devout snowBOT
@ocean scroll Has your question been resolved?
devout snowBOT
@ocean scroll Has your question been resolved?
devout snowBOT
@ocean scroll Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
boreal helm
how to determine solvability of the diophantine equation Ax^2-By^2=C generalised pell's equation
devout snowBOT
how to determine solvability of the diophantine equation Ax^2-By^2=C generalised...
@boreal helm Has your question been resolved?
devout snowBOT
@boreal helm Has your question been resolved?
mellow raft
sec θ + tan θ = p, prove that sin θ = p²−1/p²+1 how to do these kind of prove qs even if I do manage to do 1 I am unable to do other prove q
devout snowBOT
sec θ + tan θ = p, prove that sin θ = p²−1/p²+1 how to do these kind of prove qs...
woven radishBOT
Ann
mellow raft
I'll send a screenshot no this isn't thad
pseudo basin
ok then you definitely forgor brackets
mellow raft
pseudo basin
ok then that's properly written as (p^2 - 1)/(p^2 + 1)
anyway, most questions like this end up with like, applications of pythagoras' identity and other trig identities like angle sum or double angle
mellow raft
ok then that's properly written as (p^2 - 1)/(p^2 + 1)
Oh ok
pseudo basin
in particular here i would start with $$\frac{(\sec(\theta)+\tan(\theta))^2-1}{(\sec(\theta)+\tan(\theta))^2+1}$$ and see how well that thing yields to simplification
woven radishBOT
Ann
pseudo basin
writing all functions in terms of sin and cos also generally doesn't hurt
mellow raft
Ok noted
Wait I'm trying to solve it I'm writing in my notebook
It looks like a minus b by a plus b so should I know first try to simplify it or multiply such that I can use a² minus b²
pseudo basin
your handwriting 💔
tender wharf
It looks like a minus b by a plus b so should I know first try to simplify it or...
$(a+b)^2=a^2 + b^2 + 2ab$
pseudo basin
It looks like a minus b by a plus b so should I know first try to simplify it or...
also you expanded the square wrong
woven radishBOT
oppenheimer
pseudo basin
$(\sec(x)+\tan(x))^2 \neq \sec^2(x)+\tan^2(x)$
woven radishBOT
Ann
mellow raft
After simplifying i got 1/sin² + sin² by sin² -1/sin² ...oh mb let me do it again
Sorry
pseudo basin
also you managed to make the n completely not look like n
it looks more like a sloppy z or 2
warped fulcrum
When you redo the question, I suggest to multiply the top and bottom by cos²θ
mellow raft
Wait what wrong did I do
Other than the last step even if I did it right now will it become sin theta
warped fulcrum
Can you type out what you have in the numerator and denominator
pseudo basin
this step, how did it happen / what did you attempt to do
also i will keep saying that your handwriting could use a lot of improvement
warped fulcrum
this step, how did it happen / what did you _attempt_ to do
Where did the 2 go?
pseudo basin
got cancelled out
that isnt the issue
op had (2tan^2(t) + 2sec(t)tan(t))/(2sec^2(t) + 2sec(t)tan(t))
also he's suppressing the argument on all trig functions but. whatever ig
mellow raft
In numerator I have sin²/cos² plus (1+sin)/cos² in denominator 1/cos²+(1+sin)/cos²
pseudo basin
it sounds like you tried to turn $$\frac{\frac{a_1}{b_1} + \frac{a_2}{b_2}}{\frac{a_3}{b_3} + \frac{a_4}{b_4}}$$ into $$\frac{a_1}{b_1} + \paren{\frac{a_2}{b_2} \times \frac{b_3}{a_3}} + \frac{b_4}{a_4}$$
woven radishBOT
Ann
mellow raft
Yes
mellow raft
Why
pseudo basin
1/(a+b) ≠ 1/a + 1/b, and missing brackets
pseudo basin
wtf went on in the last step
mellow raft
Sin common
pseudo basin
also this is dubious anyway
actually even the first nested fraction in your thing is causing me doubts
you're asserting sec(x)tan(x) = (1+sin(x))/cos^2(x) ?
$\sec(x) \tan(x) \neq \frac{1+\sin(x)}{\cos^2(x)}$.
woven radishBOT
Ann
mellow raft
But 1/cos into sin/cos wait
...... It will be sin/cos² right?
pseudo basin
yes the numerator is going to be just sin(x), not 1+sin(x).
so then if that's fixed, you end up with not $\frac{1+\sin(x)+\sin^2(x)}{1+1+\sin^2(x)}$ but actually the correct fraction would be $\frac{\sin(x)+\sin^2(x)}{1+\sin(x)}$.
woven radishBOT
Ann
mellow raft
Hmm now I can't think of how to solve it further by simplifying so should I divide numerator and denominator by cos?
That didn't help
What is there that i can do
I have to get sin theta so I must cancel denominator and get sin in numerator
pseudo basin
sin(x) + sin^2(x) can be factorized
devout snowBOT
@mellow raft Has your question been resolved?
proud perch
What happens to the quadratic formula under permutations of coefficients?
proud perch
The swap of the quadratic and constant coefficients yields the root map x -> 1/x but I'm curious to see if there's something else
polar chasm
i dont think there is going to be anything nice
a and c are kinda symmetric in the formula, the only difference is /2a
but b^2 - ac is completely symmetric in a,c
on the other hand, b isnt symmetric with anything
shadow bear
Richard has a four infinitely large piles of coins: a pile of pennies, a pile of nickels, a pile of dimes, and a pile of quarters. He chooses one pile at random and takes one coin from that pile. Richard then repeats this process until the sum of the values of the coins he has taken is an integer number of dollars. What is the expected value of this final sum of money, in cents?
devout snowBOT
Richard has a four infinitely large piles of coins: a pile of pennies, a pile of...
hollow jolt
whats the expected value from 1 trial
devout snowBOT
@shadow bear Has your question been resolved?
shadow bear
10.25?
shadow bear
helpp
vital edge
,calc (1+5+10+25)/4
woven radishBOT
Result:
10.25
Yeah it would be that
Well how many times do you have to add 10.25 cents to get an integer number of dollars
Wait I'm so confused now that I think about it more
Do you have an image of the original question
shadow bear
yeah
1sec
its basically the same but with the source that i cant find ee
eh wait
wrong
shadow bear
ok
silk rock
why do I feel like the answer would be... rather surprising and not surprising at the same time
silk rock
yea I shoved linearity of expectations away but good to know
vital edge
Yeah this is more like multiplying expectations
So we care about two things
- How much the sum is expected to increase every step
- What is the expected number of steps
We have already done (1)
Any ideas how to do (2)? @shadow bear
shadow bear
lemme tink
let P(x) be the probability that we finish the game in x steps, and E(x) the expected value of the final number of coins after finishing the game in x steps, then (2) = sum(P(x)E(x))? since sum(P(x)) = 1
vital edge
As a hint, 2 is unrelated to 1
silk rock
I have a feeling that you're right but at the same time the way E(x) is worded makes me not confident in stamping the seal of approval
shadow bear
uh
i mean if we can finish the game in x steps, then E(x) is the expected value of the number of coins if we finish the game in x steps, otherwise it's 0
shadow bear
is it useful? like is there a general formula for P and E
P(4) = 1/4^4, hmm
but then it might be kinda complicated cuz P(k) is related to P(k-1)
actually P(k+1) = (1-P(k)) * something
or more like (1-P(1)(1-P(2)...(1-P(k))
multiplied by something
vital edge
Are you familiar with the idea of a random walk
vital edge
Well we want the final sum to be an integer
And we are adding 0.01, 0.05, 0.10 and 0.25 at every step
Let's multiply by 100 to make it simpler
We want the final sum to be 0 mod 100
And we are adding 1, 5, 10 or 25
shadow bear
mm yea
vital edge
What's the expected number of steps to come back to zero
vital edge
Well how do we find the number of steps in a random walk
shadow bear
uhh by looking at its deviation?
i only know the random walk where one has a 1/2 probability to go up or down by 1 unit in the cartesian plane every step
silk rock
consider the number of possible states in this walk
shadow bear
100
silk rock
I'm sure that Xavier would delete me, but I'm also pretty sure this walk is finite and irreducible
shadow bear
💀
silk rock
on top of that, the probability of the next coin doesn't depend on the current state
now how to properly put this in a way that makes sense, I will need Xavier's help
vital edge
Mika I have never done random walks in my life
I just have enough intuition to intuit my way through it
Also I just realised I was picturing this as a markov chain lmfao
silk rock
and it is!
vital edge
I know, but that probably isn't very helpful to the helpee
silk rock
fair but if so I am not sure how to explain why this will result in a uniform distr. over all states
vital edge
True
I'm gonna wait for someone more skilled at probability to walk in here cuz this needs a lot more skill than either of us have
copper harbor
The markov chain is probably the reasonable way to approach this
You have 101 states, and I don't think the system you have to solve should be too terrible to solve
silk rock
You have 101 states, and I don't think the system you have to solve should be to...
isn't it 100? just double-checking
copper harbor
isn't it 100? just double-checking
hm the system I have you have a state for each cent value from 1 to 100, and a "starting state"
The reason for the starting state is you don't want 0c at the start to be an absorbative state
silk rock
ah alright, noted
devout snowBOT
@shadow bear Has your question been resolved?
ebon coyote
As in, the 0c shouldn't be dependent on "previous" states, because there's no such previous state in the first place
I presume
At a risk of it being a nosols, the sheet you mentioned does contain this solution sketch:
Only, I'm not sure I agree with the sketch, since I'd argue the subscripts there should be n-1, n-5_, etc.
Not that I'm that versed in this either, but if you then have the right formulation, with some justification ig, then this question boils down to "how many of these EVs equal zero?"
ebon coyote
I love the "Do algebra." that comes after that 
But yh, justify this formula (or whichever formula ig, because I personally differ on this), then... idk do algebra
shadow bear
At a risk of it being a nosols, the sheet you mentioned does contain this soluti...
wait but i dont understand why this is true
hm ok
oh wait x_n is not like x_i in the sense of a sequence
ebon coyote
Functionally, I'd posit that it's because being at a state of x_n either involves you having picked up a 1c coin while in x_(n-1) (a 1 in 4 chance of happening) or a 5c coin while in x_(n-5) (a 1 in 4 chance again), etc.
yh
^ this also explains why I differ from this sketch; only, the sketch itself here doesn't really tell you why it's written as it is
shadow bear
wait uh i judt did some googling and found walds equation
is it useful here?
altho i cant seem to find E(N), expected value of number of steps to win
devout snowBOT
@shadow bear Has your question been resolved?
sudden crater
I have another analysis question - I need to prove or disprove this statement:
There exist two convergent series $\sum_{n=1}^{\infty}a_{n}$ and $\sum_{n=1}^{\infty}b_{n}$ such that $b_{n}$ is obtained by rearranging $a_{n}$'s terms, and:
$\sum_{n=1}^{\infty}b_{n} = \pi + \sum_{n=1}^{\infty}a_{n}$
and:
$\sum_{n=1}^{\infty}|b_{n}| = \sqrt{ 3 }\cdot \sum_{n=1}^{\infty}a_{n}$
devout snowBOT
I have another analysis question - I need to prove or disprove this statement:
T...
woven radishBOT
steveRoll
sudden crater
pi and sqrt 3 obviously look like dummy values, and I think we're expected to use this theorem that we have:
Let $\sum a_{n}$ be a known series and $\sum b_{n}$ a series that is obtained by rearranging its terms.
$\sum a_{n}$ is absolutely convergent if and only if $\sum b_{n}$ is absolutely convergent. When they are absolutely convergent, their sums are equal.
woven radishBOT
steveRoll
cerulean finch
yes this theorem basically gives you all the information to prove, or disprove, such statement
devout snowBOT
@sudden crater Has your question been resolved?
sudden crater
yes this theorem basically gives you all the information to prove, or disprove, ...
I can't see it right off the bat 😭
I can't even see how rearranging a series' terms would change its sum at all
toxic flower
I can't even see how rearranging a series' terms would change its sum at all
well.. it could change if series is conditionally convergent
sudden crater
but how
toxic flower
$$S_1 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots $$
woven radishBOT
Shikhar
toxic flower
$$S_2 = 1 - \frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{6} - \frac{1}{8} + \frac{1}{5} - \frac{1}{10} - \frac{1}{12} + \cdots$$
woven radishBOT
Shikhar
sudden crater
gotta be honest I still don't see it
young spade
void fox
The intuition is this. If a series conditionally converges then its positive and negative parts must each have a divergent sum. At the same time they must -> 0 since the series at a whole converges. Therefore by alternatingly adding positive and negative terms you can make the sum hover around any value you like
This is the Riemann rearrangement theorem
woven radishBOT
Shikhar
toxic flower
hope you understand
you can make this series approach to any number by just rearranging its terms
uncut crow
aren’t sum a_n and sum b_n both absolutely convergent anyway by the facts
- they are both convergent and b_n is absolutely convergent (last part)
- sum a_n is a rearrangement of sum b_n
sudden crater
guess I gotta research that some more... I don't see it intuitively at all
void fox
I was just explaining how rearranging a series may affect its value since somebody brought that up and they seemed confused by it
uncut crow
aren’t sum a_n and sum b_n both absolutely convergent anyway by the facts
1) th...
this plus the theorem that says sum a_n = sum b_n under these conditions completes the problem
understandable blante. carry on
void fox
thank you sleighla
uncut crow
no problem daddy blante
timber stratus
you don’t need it for ts problem
old coral
how do you know sum a_n = sum b_n?
isnt the theorem only saying that sum |b_n| = sum|a_n|?
uncut crow
they are rearrangements of each other and sum b_n is absolutely convergent
rearranging an absolutely convergent series does not change its value
toxic flower
they are rearrangements of each other and sum b_n is absolutely convergent
who the hell said they are absolutely convergent?
uncut crow
aren’t sum a_n and sum b_n both absolutely convergent anyway by the facts
1) th...
i did, right here
toxic flower
**steveRoll**
talking bout this 
uncut crow
yea, what i said is a consequence of the problem hypotheses
read the very last part
in the problem
also just btw the pi and the sqrt(3) are red herrings. you could put any nonzero numbers there and it wouldn’t really affect the problem
sudden crater
so like, with everything I gathered so far, I just know that:
- b_n is absolutely convergent (and thanks to the theorem, so is a_n, and sum |a_n| = sub |b_n|)
- but this doesn't say anything about their non-absolute sums
uncut crow
yes it does
uncut crow
**steveRoll**
we have sum a_n = sum b_n, from this, do we not?
i’m pretty sure that’s what ‘their sums are equal’ is referring to. and even if it’s not, it’s still true that they are equal
sudden crater
oh...from how it's worded, I thought it said sum |a_n| = sum |b_n|
uncut crow
when ‘they’ are absolutely convergent (so, ‘they’ must be the non absolute sums), ‘their’ sums are equal
sudden crater
huh
uncut crow
we have sum a_n = sum b_n, from this, do we not?
anyway do you see why the problem is done if we have this?
sudden crater
yeah, that would make a contradiction immediately
uncut crow
i’m pretty sure that’s what ‘their sums are equal’ is referring to. and even if ...
i’m growing more and more sure this is the intention of the problem by the second
sudden crater
.close
devout snowBOT
silent oriole
devout snowBOT
Available help channel!
night rune
is there a cunning way for this instead of long division
devout snowBOT
is there a cunning way for this instead of long division
night rune
i am not supposed to use a calculator
night rune
i solve this by splitting 14 to 7*2 .. i removed the 2 and handled N = 775 * 389 * 781
not sure if this is even possible
but 5 * 9 * 1 = 45 (units digit)
/7
remainder = 3
harsh sierra
use backslash before *
night rune
so i thought the remainder would follow a multiple of 3
i picked 6
ill try your way
devout snowBOT
@night rune Has your question been resolved?
vernal grail
hi
woven radishBOT
alee
vernal grail
where delta(n) is the kronecker delta
since this equality is true for every n
i can write
$x(\cdot) = \sum_k x(k) \delta(\cdot - k)$
oh
how can i write the dot inside ?
harsh sierra
\cdot
woven radishBOT
alee
vernal grail
if i assume that $\mathcal T$ is a linear system
woven radishBOT
alee
woven radishBOT
alee
woven radishBOT
alee
vernal grail
so
$y(n) = \mathcal T{x(\cdot)}(n) = \sum_k x(k) \mathcal T {\delta(\cdot - k)}(n) = \sum_k x(k) w(n,k)$
woven radishBOT
alee
vernal grail
so this formula is valid for any linear system right ?
and w(n,k) means that the output can depend on n and k ?
i dont understand what does this notation mean w(n,k)
i know that $\mathcal T{x(\cdot)}(n) = y(\cdot)$
woven radishBOT
alee
vernal grail
i.e. i apply the operator T to x and i get a new signal y
oh
I read that in w(n,k) k is the excitation index, what does it mean?
also they say that we can see this as
take $\ell \triangleq n - k \Longleftrightarrow k = n-\ell$
woven radishBOT
alee
vernal grail
so we get $w(n,n-\ell)$
woven radishBOT
alee
woven radishBOT
alee
woven radishBOT
alee
vernal grail
so
can someone explain the difference between w(n,k) and h(n,l) pls
i know that for example $\mathcal T{\delta(\cdot-1-k)}(n) = w(n,k+1)$
woven radishBOT
alee
vernal grail
so w(n,k) is the output at time n when the input is a delta shifted by k
so h(n,l) = w(n,k) ?
i dont understand what they are trying to say with h(n,l)
<@&286206848099549185>
is it just $\mathcal T{\delta(\cdot - \ell) }(n) = h(n,\ell)$ ?
woven radishBOT
alee
vernal grail
maybe because they mean h as impulse response ?
frigid phoenix
How do I graph this, all I know how to do is graph the vertical asymptote at x=3
hollow jolt
graph it how you would graph any function
frigid phoenix
I don't know what that means
frigid phoenix
How did you get the function x^2-2x+1
frigid phoenix
(x-1)^2
steady kettle
x+2=x-3+5
frigid phoenix
Oh ok
steady kettle
(x+2)/(x-3)=(x-3+5)/(x-3)
frigid phoenix
I don't even know how to graph that
hollow jolt
well if you give the function any x value you get an output
steady kettle
can you draw 5/x
hollow jolt
so you could just find a bunch of correlating x and y values
and plot them on the axes
and join the dots
steady kettle
ye
frigid phoenix
Ok
steady kettle
draw 5/(x-3) first
(8,1) for example
frigid phoenix
Ok so for the actual equation I could plug in 1 for both x and y and that would give me one point
frigid phoenix
Don't you have to worry about the asymtotes
hollow jolt
yes but when you plot points close enough to them you can see which is going up and which is going down
devout snowBOT
devout snowBOT
Available help channel!
mild plaza
Hey
mild plaza
Can someone explain this for me
It’s absolute value rule
I just don’t understand it
And in sets may I know what are the rules for the bars so like bar A intersect bar B what is it exactly
devout snowBOT
@mild plaza Has your question been resolved?
mild plaza
.close
devout snowBOT
devout snowBOT
Available help channel!
solar goblet
devout snowBOT
solar goblet
does $\vdash$ mean the same as $\rightarrow$?
woven radishBOT
vagrant skiff
no
vagrant skiff
varphi to psi is a formula meaning if varphi is true, then psi is true (in a given valuation/model).
Gamma vdash psi is a meta-level statement moaning something like from assumptions/premises Gamma, there is a formal proof of psi in a given proof system.
solar goblet
so it means you can transform from one premise to a conclusion?
so something like $\lnot\lnot p \vdash p$
woven radishBOT
vagrant skiff
yeah, it’s derive via inference rules
vagrant skiff
assuming you're doing classical logic, yes
devout snowBOT
devout snowBOT
Available help channel!
stray steeple
The matrix $A = \begin{bsmallmatrix}2 & 0 \\ 0 & 1\end{bsmallmatrix}$ produces a stretching in the $x$-direction. Draw the circle $x^2+y^2=1$ and sketch around it the points $(2x,y)$ that result from multiplication by $A$. What shape is that curve\?
0_א
stray steeple
The answer in the answer key is
$\norm{Ax}^2=1$ always produces an ellipse.
0_א
woven radishBOT
0_א
stray steeple
Am I doing it correctly? Because that equation gives a vertical ellipse; it doesn't stretch it but it contracts the circle to form a vertical ellipse.
But from my understanding of the question, it should stretch the circle horizontally to form a horizontal ellipse
errant harbor
Well the ellipse should actually be x^2/4 + y^2 = 1
The reason is, we're taking the points (x, y) such that x^2 + y^2 = 1,
Then we plot (u, v) = (2x, y)
stray steeple
So yea, i also wanna know if the problem is just incorrect
errant harbor
The reason is, we're taking the points (x, y) such that x^2 + y^2 = 1,
Then we ...
=> (u/2) ^2 + v^2 = 1
stray steeple
How do I get $4x^2+y^2=1$\?
\begin{align*}
\norm{\begin{bmatrix}
2 & 0 \\ 0 & 1
\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}}^2 &= 1 \\
\norm{\begin{bmatrix}2x\\y\end{bmatrix}}^2 &= 1 \\
4x^2+y^2 &= 1
\end{align*}
0_א
stray steeple
=> (u/2) ^2 + v^2 = 1
ok let me read that
stray steeple
=> (u/2) ^2 + v^2 = 1
oh ok i think i get it
errant harbor
**0\_א**
In here, u transform (x y) first and then set the length equal to 1.
You're supposed to set the length of (x y) = 1 FIRST, and then transform it by the matrix A
errant harbor
In here, u transform (x y) first and then set the length equal to 1.
You're supp...
So, u basically did it in the wrong order
stray steeple
I see. I'll work with it. Thank you!
devout snowBOT
If you are done with this channel, please mark your problem as solved by typing .close
stray steeple
.close
devout snowBOT
devout snowBOT
Available help channel!
gentle ginkgo
what do I do for -degrees
devout snowBOT
what do I do for -degrees
drifting sierra
You can add or subtract 360º as many times as you want
gentle ginkgo
You can add or subtract 360º as many times as you want
so is there somethin different I need to put in besides 1?
drifting sierra
Why would it be 1?
gentle ginkgo
Why would it be 1?
idk tbh I just saw 90 degrees with sin and just thought 1
drifting sierra
... so you just didn't read what I wrote
warped fulcrum
Wait, have you heard of something called ASTC
gentle ginkgo
Wait, have you heard of something called ASTC
Nope
gentle ginkgo
... so you just didn't read what I wrote
how would I put that into what I'm doing
drifting sierra
Add or subtract 360º as many times as you need to -90º so that it's in the range 0º to 360º
small inlet
sin(-90) = 2 tho
Least believable ragebait
umbral snow
what
drifting sierra
Ah yes, take the first answer you get without understanding it and complain it's incorrect
warped fulcrum
Wait, have you heard of something called ASTC
This is ASTC
small inlet
and my answer is wrong
If you were to add a full circle = 360 deg to -90 deg you would have the equivalent expression which is sin(?)
warped fulcrum
This is ASTC
It refers to the sign of the basic 3 trigo functions based on how many degrees
gentle ginkgo
It refers to the sign of the basic 3 trigo functions based on how many degrees
I'm going to right this down
gentle ginkgo
If you were to add a full circle = 360 deg to -90 deg you would have the equival...
wait a minute
it would take me to 270 degrees and since i'm looking for sin it would be -1?
holy moly my brain is growing
I think I got this
.close
devout snowBOT
gentle ginkgo
.reopen
devout snowBOT
✅ Original question: #help-27 message
small inlet
cos(-x)=cos(x)
gentle ginkgo
is there any other rule I should be aware of?
and also how would I find the tan if they ask for it
warped fulcrum
Always rely on ASTC
If its not within the range, just add or subtract 2pi radians or 360°
gentle ginkgo
If its not within the range, just add or subtract 2pi radians or 360°
not to sound like a complete fool but what does within range mean
warped fulcrum
Like the base is from 0° to 360°
devout snowBOT
devout snowBOT
Available help channel!
thorny rapids
Can anyone explain the pigeonhole principle? - Combinatorics
devout snowBOT
Can anyone explain the pigeonhole principle? - Combinatorics
void fox
The pigeonhole principle just says there can be no injection from a set of larger cardinality to a set of strictly smaller cardinality
thorny rapids
The pigeonhole principle just says there can be no injection from a set of large...
Simpler terms please....
void fox
If you have m items and you put them into n buckets, m>n, then there is at least one bucket with at least 2 items
thorny rapids
If you have m items and you put them into n buckets, m>n, then there is at least...
Alright But how do I use this in questions
void fox
Depends on the question
thorny rapids
Depends on the question
Any type of question...Th emost basic one you can think off
uncut crow
Any type of question...Th emost basic one you can think off
there are 10 integers in front of you. prove there are two them that have the same remainder when divided by 9
thorny rapids
there are 10 integers in front of you. prove there are two them that have the sa...
Since there are 10 Pigeons (Intergers) > 9 Pigeonhouse (Remainders of 9 1-9)
so at least two numbers will share the same interger?
Is this how you do it
granite venture
there are 10 integers in front of you. prove there are two them that have the sa...
the least basic i could have thought of
oh nvm
i somehow thought this was the question about numbers with only 1s
uncut crow
Since there are 10 Pigeons (Intergers) > 9 Pigeonhouse (Remainders of 9 1-9)
so...
yes but i would call the remainders 0, 1, …, 8
granite venture
Since there are 10 Pigeons (Intergers) > 9 Pigeonhouse (Remainders of 9 1-9)
so...
thats the idea yeah
calling the remainders 1 through 9 is okay but unconventional
thorny rapids
So how would you do a question where
A die is rolled a set amount of time
What is the smallest amount of rolls to get the same number twice
thorny rapids
"
A die is thrown repeatedly.
a. What is the smallest number of times the die must be thrown to ensure at least one number turns up twice"
thorny rapids
6
granite venture
how many rolls until its not possible to get that all of them are different
thorny rapids
7 rolls
granite venture
so the answer is...
thorny rapids
7 because it would have to repeat a number
granite venture
yeah
thorny rapids
What if it was specific and asking 6
granite venture
explain
thorny rapids
What is the smallest number of times the die must be thrown to ensure that a six is rolled twice" - So same answer?
thorny rapids
Is it because you cant find a specific number but just a generalised?
thorny rapids
its possible to roll numbers 1 to 5 endlessly
alright
Do you have any good resources to practise this
granite venture
the pigeonhole principle is only about something happening twice, not which thing it is
thorny rapids
the pigeonhole principle is only about something happening twice, not which thin...
alright
granite venture
if you saw the moon 13 times in a year, then two of them were for sure in the same month
but that month does not have to be january
thorny rapids
So basically you cant find a specific thing only a generalised thing
granite venture
yeah
you could have had 3 moon sightings in february and none in january
but it is impossible to have only one moon sighting in each month for a total of 13 moon sightings
devout snowBOT
@thorny rapids Has your question been resolved?
thorny rapids
How would you do this type of question?
devout snowBOT
How would you do this type of question?
thorny rapids
I( know the total is (4C2x 4C2 x 2) (4C2*4C2 x 2) / 2! How would do this restriction?
devout snowBOT
@thorny rapids Has your question been resolved?
thorny rapids
did you mean to type (4C2x 4C2 x 2) twice?
Ment to be (4C2 * 4C2 * 2)(2C2 * 2C2 * 2) / 2!
alpine python
so 72? it's still too many, i think
brittle coral
Ment to be (4C2 * 4C2 * 2)(2C2 * 2C2 * 2) / 2!
maybe explain your working so we can see your reasoning
alpine python
oh, 72 is the number of games if you allow a husband and wife to play together
maybe that's what you meant
red grove
Yup, but the formula should not have this form
(If you’re searching for the number of games allowing husband and wife, I mean)
alpine python
i've found that with counting, there's usually different ways to get to the same answer
brittle coral
How would you do this type of question?
hmm looks like you have to do cases because these cases are clearly different:
Case 1:
M1 W2
M2 W?
Case 2:
M1 W2
M(3/4) W?
red grove
i've found that with counting, there's usually different ways to get to the same...
Definitely
brittle coral
i've found that with counting, there's usually different ways to get to the same...
usually there is either one uniform procedure or you need special cases
alpine python
hmm looks like you have to do cases because these cases are clearly different:
...
this is one approach. you can also try to subtract the cases where couples play together from the total 72
i think tom tom's approach is easier here
brittle coral
this is one approach. you can also try to subtract the cases where couples play ...
just tried this in my head and this requires PIE 😭
thorny rapids
maybe explain your working so we can see your reasoning
Thats the total, basiccally just choose 2 men and 2 women for 1 game. *2 because there is two different cases either you have the 1st women or the 2nd women.
I tried to do math gpt -
it did Total - (Case A + Case B) didnt understand the cases
alpine python
just tried this in my head and this requires PIE 😭
inclusion-exclusion?
brittle coral
I tried to do math gpt -
it did Total - (Case A + Case B) didnt understand the ...
that might be axe's suggested approach
alpine python
yeah you have to consider that one, or both teams could consist of a couple
thorny rapids
What will the two Cases be tho?
brittle coral
I tried to do math gpt -
it did Total - (Case A + Case B) didnt understand the ...
wait a minute
Total - (A + B)
??
ok that's definitely not PIE
brittle coral
What will the two Cases be tho?
might be
exactly 1 pair couple
2 pairs couples
but the exactly 1 pair couple case is irritating to count
alpine python
it's doable
alpine python
exactly 1 pair is || 24 ||
thorny rapids
thorny rapids
Holdup
brittle coral
also here we dont advocate using chatgpt for math
thorny rapids
thorny rapids
also here we dont advocate using chatgpt for math
mb i find it helpful - somewhat
alpine python
24 and 6 looks right
devout snowBOT
devout snowBOT
Available help channel!
tranquil steppe
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a coastal access road. He drove three times as fast on the highway as on the coastal road. If Anh spent 30 minutes driving on the coastal road, how many minutes did his entire trip take?
$\textbf{(A) }50\qquad\textbf{(B) }70\qquad\textbf{(C) }80\qquad\textbf{(D) }90\qquad \textbf{(E) }100$
devout snowBOT
On a trip to the beach, Anh traveled 50 miles on the highway and 10 miles on a c...
woven radishBOT
yippeeeeeeeeeee
tranquil steppe
I saw it but my dumbahh came up with 330
eager scaffold
so given:
he's 3x faster on coastal road than on the highway
he spent 30mins on coastal road
tranquil steppe
so I tried to make equations
tranquil steppe
1/3 speed right
eager scaffold
yep
eager scaffold
oh yea mb
so if he's 3x faster on highway
then he spends 3 times less time on that road
meaning this time 10 miles = 10 minutes
tranquil steppe
I also came up with 60...
eager scaffold
hold on
eager scaffold
50+30
devout snowBOT
grand edge
devout snowBOT
Available help channel!
civic stump
i am learning cubic functions and i dont really understand how it works. i found out that some of them (like the picture on the right) tend to have fluctuations bewteen the lines going up and down. i also realized that the difference happens in some scenarios where x powered by 2 is used too, but it doesnt always work that way. i also dont know how to find the extreme points (sorry for possibly bad english, i am not a native speaker)
devout snowBOT
i am learning cubic functions and i dont really understand how it works. i found...
civic stump
(the function i used for the graph on the right is x³+2x²)
supple knot
i am learning cubic functions and i dont really understand how it works. i found...
the easiest way to learn this is with calculus, so first learn limits and derivatives
or if you just want the answers without justification:
https://www.savemyexams.com/igcse/maths/cie/international-maths/23/extended/revision-notes/coordinate-geometry-and-graphs/types-of-graphs/cubic-graphs/
civic stump
but maybe you can explain what equation or formula is used to find the extreme point
like if you solve it in front of me/with me i would learn it better than digging stuff up from google
round raptor
but maybe you can explain what equation or formula is used to find the extreme p...
do you know what extreme points are, also do you know differentiation ?
civic stump
its familiar but im unsure
round raptor
hm.. what is your native language ?
civic stump
estonian
round raptor
ok
supple knot
civic stump
but dont worry i dont know what it means even in my language😭
so really i suppose my issue is i cant do differentiation
round raptor
but dont worry i dont know what it means even in my language😭
oh have you done any calculus/precalc
civic stump
uhm well..
supple knot
the easiest way to learn this is with calculus, so first learn limits and deriva...
yea you just gotta learn limits and derivatives then
civic stump
i didnt really pay attention so now i dont have the foundation
so im just learning from the hardest point, trying to get down to the basics
round raptor
then i would say working that first
round raptor
also you can try organic chemistry tutor on youtube
devout snowBOT
Available help channel!
glossy dew
hrmp
glossy dew
i have a solution and im struggling to make sense of it from step 3
devout snowBOT
@glossy dew Has your question been resolved?
supple knot
they used log(a) + log(b) = log(ab)
round raptor
i have a solution and im struggling to make sense of it from step 3
What is step 3
devout snowBOT
devout snowBOT
Available help channel!
errant sierra
Given a geometric series a_1, a_2, a_3... whose terms are all positive and whose quotient is q.
A new series b is constructed so that
$B_n = \sqrt(a_n*a_n+1)$
Prove that the new series B is a geometric sequence
woven radishBOT
Doctor beak🎗
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
hollow jolt
yeah youre good so far
toxic flower
Given a geometric series $a_1, a_2, a_3$... whose terms are all positive and whose quotient is q.
A new series b is constructed so that
$B_n = \sqrt{a_n*a_{n+1}}$
woven radishBOT
Shikhar
toxic flower
you mean this?
errant sierra
Yes
errant sierra
yeah youre good so far
So would the quotient of B be q^2?
Or q^3?
I mean, I could rewrite $\frac{\sqrt{q^3}}{\sqrt{q}} as \frac{q^{3.5}}{q^{0.5}}$
woven radishBOT
Doctor beak🎗
sand dove
Sqrt(q³) is not q^(3.5)
errant sierra
Is it, like, (q^3)^0.5?
Then?
Oh thanks, I understand
Yeah
Thanks
The quotient is q
.close
devout snowBOT
devout snowBOT
Available help channel!
pearl grotto
devout snowBOT
pearl grotto
why here if we take the moment around b
then the normal reaction force disappears, but 60N still remains?
although that force is also acting at B?
devout snowBOT
@pearl grotto Has your question been resolved?
devout snowBOT
@pearl grotto Has your question been resolved?
ocean scroll
If you take the moment at B, the normal reaction force stays, but the 60N disappears
Because the normal reaction force is at A
The 60N force is at B so it's 0 distance away from B, so it doesn't contribute
Do you have a diagram of all the forces with the answer for 11a?
wicked turtle
sir, this is a math server
neon wagon
can someone help with the b) im completely lost on thah
devout snowBOT
can someone help with the b) im completely lost on thah
neon wagon
think what the 4 means too
uhh
neon wagon
why is there one (2/5) and three (3/5)at the other
thats for the green ball
the (3/5)^3 is that 3 kids get non green ballss
steady kettle
and..
neon wagon
idk about the 4
neon wagon
yeah
steady kettle
how many possibilities?
steady kettle
there it goes
steady kettle
the others are for...
steady kettle
lets say the 4 child is ABCD
i think i was too fast for the meaning of 4
ill say G for green
N for non green
the kids are ABCD
steady kettle
ABCD
GNNN
NGNN
NNGN
NNNG
lets look at the first one
A gets green
and the other 3 gets non green
how's that possibility
steady kettle
A gets green
(2/5)
steady kettle
and the other 3 gets non green
(3/5)³
steady kettle
bcd is 0,75
(0.6)³
steady kettle
the chance to get a non green ball is?
steady kettle
3/5 is the chance of not getting a green ball
.
steady kettle
ABCD
GNNN
NGNN
NNGN
NNNG
look again
steady kettle
its choosing one kid to be green out of 4
there are 4 possibilities
A gets green or B gets green.... D gets green
steady kettle
and the others get non green
steady kettle
3 child getting a non green ball
neon wagon
3 child getting a non green ball
not 4?
steady kettle
oops 4
neon wagon
but like why add that to the other part
steady kettle
there are 2 possibilities added
all 4 getting a non green ball
only 3 getting a non green ball
how can we make it to a sentence
neon wagon
wait im thinking
neon wagon
||use 'at least'||
Out of 4 kids atleast 1 ball is green or none is green
steady kettle
||at least 3 has...?||
||this means 3 or 4 so... what did we do before?||
neon wagon
||at least 3 has...?||
at least 3 non green balls
steady kettle
YESSSsS
neon wagon
wait lowkey it makes sense tho
steady kettle
MAKE THAT INTO A SENTENCE RIGHT NOW
neon wagon
MAKE THAT INTO A SENTENCE RIGHT NOW
Whats the probability that out of 4 kids atleast 3 kids get non green balls
steady kettle
thats it
neon wagon
im still confused by the + part tho
like why is there (3/5)^4
and then times 4 again
after
steady kettle
because al least 3 gets non green ball means
the sum of possibilities that only 3 has ng ball and all 4 has ng ball
do you know nCr
steady kettle
combination
neon wagon
yeah but im rly bad at it
steady kettle
that means the 4 at the front
steady kettle
chance that A gets green is
(3/5)³×(2/5)
chance that B...
C.. D...
neon wagon
but the (3/5)^4 dont include the 4?
neon wagon
chance that A gets green is
(3/5)³×(2/5)
chance that B...
C.. D...
yeah
steady kettle
but the (3/5)^4 dont include the 4?
because all 4 gets non green balls
same as choosing 4 out of 4
steady kettle
ABCD
GNNN
NGNN
NNGN
NNNG
there was four here
neon wagon
because all 4 gets non green balls
oh
steady kettle
got it?
neon wagon
yeah ty
devout snowBOT
devout snowBOT
Available help channel!
lime osprey
i need help with this question (inconsistent just means the system has no solutions)
devout snowBOT
i need help with this question (inconsistent just means the system has no soluti...
lime osprey
idk how to start it so i dont have work
ripe grove
What does inconsistent have to do with the determinant...
pearl relic
i need help with this question (inconsistent just means the system has no soluti...
inconsistent means one is the multiple of the other
lime osprey
oh
pearl relic
and if u add them both u get 0=5 or something like that
so what can you multiply by -9x+9y to get -8x+(something)y
pearl relic
oh
sorry no it means both you're still right
if one is a multiple of the other then the system has no solutions
pearl relic
What does inconsistent have to do with the determinant...
somewhat doubt that this is a lin alg problem
if it is then you could also do that
ripe grove
A system doesn't have to have linear dependency to be inconsistent
lime osprey
it is in my lin alg course but its the first hw (idk how to find determinants yet lol)
lime osprey
so what can you multiply by -9x+9y to get -8x+(something)y
wait what
ripe grove
You could do the very primitive thing here then and just solve for y for both equations and equate their gradients
lime osprey
-9x to -8x
what do you multiply by
8/9
pearl relic
8/9
ok now you do the same thing with the 9y to get ky
multiply 9y by 8/9 to get ky is 8y
so k is 8
try solving the system when k is 8 to see what i mean
lime osprey
does that always work
pearl relic
does that always work
well its either inconsistent or infinitely many solutions
you just have to check every time
lime osprey
wait lemme graph it i think im conceptualizing it wrong
pearl relic
yeah that sounds like a good idea
lime osprey
ohh i get it know yeah
pearl relic
oh btw
lime osprey
for some reason i thought the right hand sides had to be multiples of each other too
pearl relic
for some reason i thought the right hand sides had to be multiples of each other...
that would mean infinitely many solutions
for an inconsistent system that cant happen
lime osprey
that would mean infinitely many solutions
yeah
pearl relic
this only works if both equations look like Ax+By=C btw
well
actually you can do it in any form really
you just figure out the multiple and do it on each term
pliant crescent
for some reason i thought the right hand sides had to be multiples of each other...
if theyre multiples of eachother u have infinite solutions
lime osprey
you just figure out the multiple and do it on each term
ok ok got it
thanks for ur help!
pliant crescent
this is for a linalg course?
pliant crescent
u might want to hone your algebra skills
lime osprey
true
its been a while since i took my last algebra course
need to learn a few things again haha
.close
devout snowBOT
devout snowBOT
Available help channel!
glossy dew
uh yeah sorry for abandoning the channel 🪦
devout snowBOT
uh yeah sorry for abandoning the channel 🪦
glossy dew
from this to this
im guessing you add x^3 and (pi-x)^3 but im not sure where they got the ln(sqrt(2)) from
the answers do match so it doesnt seem to be a typo
devout snowBOT
@glossy dew Has your question been resolved?