Sn maps to Sn+1
#help-39
wet osprey
Yeah
You need to find such a 𝜙 that is injective
Then make an example in S3
Then inject it into S4
lost steppe
wait thats to show its homomorphic, to show injective then like 𝜙(a) = 𝜙(b) means a=b
wet osprey
Then go oh this example is also non abelian
lost steppe
Then go oh this example is also non abelian
ah yea
wet osprey
If you pick 𝜙 cleverly you can then use induction for Sn
If you just pick some fuckass 𝜙 then it’ll suck lmao
Cos you’ll need to find new examples every time
lost steppe
😭 i would probably end up picking the latter
wet osprey
Cos the example you pick need not work with the 𝜙
wet osprey
😭 i would probably end up picking the latter
Nah I’m sure you’ll see the pattern when yoy try it
pearl pondBOT
@lost steppe Has your question been resolved?
stoic walrus
Hey my professor assigned this I really need help!!
pearl pondBOT
Hey my professor assigned this I really need help!!
@stoic walrus Has your question been resolved?
stoic walrus
hm i guess nobody here is good enough...
plush bramble
!done
pearl pondBOT
If you are done with this channel, please mark your problem as solved by typing .close
stoic walrus
welp...
somber adder
hm i guess nobody here is good enough...
you might have better luck in the topic channels
plush bramble
hm not even tried huh
!done
pearl pondBOT
hm not even tried huh
If you are done with this channel, please mark your problem as solved by typing .close
stoic walrus
rusty with integrals huh @plush bramble ?
couldnt be me
tsk tsk tsk
i guess you are not smart enough to help others
pearl pondBOT
i guess you are not smart enough to help others
Helpers are just people volunteering their time to help you. Be polite and patient.
versed mica
i guess you are not smart enough to help others
cyan sail
i guess you are not smart enough to help others
who does bro think he is
sleek sandal
i guess you are not smart enough to help others
man ur question ain't even that hard and u need help with it
id really do that shit in my sleep
pearl pondBOT
@stoic walrus Has your question been resolved?
sleek sandal
the ans is e1,e2,e3
pearl pondBOT
the ans is e1,e2,e3
sleek sandal
to make sure im getting this right, u can real quick answer with e1,e2,e3 because it doesn't mention that vectors x1,x2,x3 need to span the subspace?
they just need to have the same span though
frozen bluff
yes you can answer with e1, e2, e3 only if the three given vectors x1, x2, x3 are linearly independent (i.e. they span all of R^3)
sleek sandal
Ahhhhhh
So if im given 4 Lin ind vectors then the answer is just the 4 elementary vectors
frozen bluff
yup exactly if you’re given 4 linearly independent vectors in R^4 they span all of R^4, so the orthonormal basis for that span would just be e1 e2 e3 e4
number of independent vectors = dimension of the space so answer is the standard basis of that dimension
pearl pondBOT
@sleek sandal Has your question been resolved?
solar gyro
pearl pondBOT
solar gyro
For this q here, my teacher worked it out in class but didn’t support his answer graphically
Is that part really necessary?
pearl pondBOT
@solar gyro Has your question been resolved?
light helm
if the question asks you to do something specific, you should
solar gyro
if the question asks you to do something specific, you should
I don’t understand how I would, though
For this problem
pearl pondBOT
@solar gyro Has your question been resolved?
pearl pondBOT
@solar gyro Has your question been resolved?
solar gyro
.close
pearl pondBOT
pearl pondBOT
Available help channel!
void grail
why √ab is that line?
pearl pondBOT
why √ab is that line?
serene sentinel
hm?
void grail
because of thales theorem?
frozen bluff
because of thales theorem?
yeah
void grail
√ab looks like sinθ also
serene sentinel
but that side's not the hypotenuse right?
you could equate sine yeah, but calculation
random ermine
the two marked angles are equal
yeah those 2 triangles are similar
so you could just use simlarity of triangles
void grail
this is a nice way to see am-gm tho
Yes, this is why I am learning it also, this is nice too, and perhaps can be combined:
random ermine
proof without words 🗿
void grail
this is a nice way to see am-gm tho
this is from Hersh, Reuben What is Mathematics Really
I wonder what is the logic of the diagram only proofs, constructivism?
edge + compass you construct, and that's the basis of constructivist logic?
void grail
I got these tiling patterns when thinking about the Cantor set generated via logistic function with μ>4, does the Cantor set has this pattern or not really?
void grail
I just find this an interesting exploration so I wonder how to explore this further to understand Cantor set and logistic function
pearl pondBOT
@void grail Has your question been resolved?
void grail
I'm learning to prove S : Λ ⟶ ∑₂ is homeo and topological conjugacy.
void grail
where Λ is Cantor set via logistic function with μ > 4
I'm trying to figure out what S is
f is the logistic function I think,
S ∘ f is itenerary after f ?
σ is shift function so σ ∘ S is shift after itinerary
snow sail
Someone might respond but you should know this might be better suited for the advanced channels 
void grail
Someone might respond but you should know this might be better suited for the ad...
thanks! good to know, and forum is better for this also?
snow sail
I don't really use the forum. I just know people into advanced topics don't usually spend a lot of time in these channels. Some do though.
pearl pondBOT
@void grail Has your question been resolved?
sleek sandal
true or false
pearl pondBOT
true or false
toxic lichen
!xy
pearl pondBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
sleek sandal
I can't tell if it's true or false because i feel like the inequality has to be <=
but at same time they said y isn't equal to least square solution so maybe it is just true right? im leaning towards true
toxic lichen
what's $\hat{x}$?
jolly parrotBOT
Ann
sleek sandal
thats the least square solution
but im looking at the question, they never said A was lin ind or lin dep columns
toxic lichen
can you send the entire q to us tho
plush bramble
but im looking at the question, they never said A was lin ind or lin dep columns
you need to give entire context like ann said
sleek sandal
toxic lichen
ok then yeah there is no default assumption of LI on cols of A
if the cols of A are lindep then yeah you dont get strict ineq in b
plush bramble
<@&268886789983436800> spam
pearl pondBOT
pearl pondBOT
Available help channel!
shadow quarry
Is anyone here good with Newtons Optimization? I'm trying to find the minimizer of a function using Newtons with a Hessian matrix, but my initial guess is pushing me towards a max. Is there any way for me to add noise to the initial guess without changing it (ie like, adding noise only if the Hessian is pointing me towards a max). I can attach my code below
plush bramble
should try #computing-software instead
shadow quarry
🫡 ill leave the original question here since it is technically math related, i can translate the math into code if i get an answer :p
but i willa lso post there if thats ok
here's a better visual of the problem
so my initial guess is currently pushing me towards a max instead of a min
plush bramble
I think I’m the only one here who’s in Special education classes 😢
stop spamming help channels
wispy helm
I’m not
wispy helm
I text 2 channels Yo and now I’m spamming
plush bramble
yea. stop
spark rune
shadow quarry
???? 😭
pearl pondBOT
@shadow quarry Has your question been resolved?
pearl pondBOT
midnight haven
i need to find the nth derivative
i've computed like the first 5
I cannnot see the pattern
in order to make a conjecture
plush bramble
use geometric series
midnight haven
use geometric series
wdym
plush bramble
$\frac{1}{1 - r} = 1 + r + r^2 + r^3 + \ldots$
jolly parrotBOT
riemann
plush bramble
have you learned that yet
plush bramble
use geometric series
.
midnight haven
formula
what did you take r as
oh
wait no
i get you
i get it
hm
but then
midnight haven
$\frac{1}{1 - r} = 1 + r + r^2 + r^3 + \ldots$
this doesn't seem right i don't think i'm getting what you're trying to do
plush bramble
these two expressions are not equal
plush bramble
yes you showed that already
midnight haven
okay
midnight haven
these two expressions are not equal
hmmm
plush bramble
yes i understood that as the nth derivative
plush bramble
**riemann**
the right side has ... do you know what that means?
midnight haven
these two expressions are not equal
isn't that the derivative?
midnight haven
the right side has ... do you know what that means?
well that's the formula for the sum of a geometric sequence
plush bramble
the bottom is the derivative of the top, correct
plush bramble
you just cut off your derivative operator in the screenshot
midnight haven
you just cut off your derivative operator in the screenshot
oh my bad !
omg what am i doing let me fix it
wait
i'll need the nth derivative of a x^n then
hm
plush bramble
you can test out a few small numbers like n=3,4,5 and find a pattern
midnight haven
you can test out a few small numbers like n=3,4,5 and find a pattern
i
hm
found this
not sure if it's right
let me plug it in the original thing
n' is the number of times of deriving
i split it from n because they might not be equal ig
Uhgggggggggg
plush bramble
i don't know what you're doing
feel free to explain what you're calculating and avoid vague words like "thing" and "that"
plush bramble
you can test out a few small numbers like n=3,4,5 and find a pattern
and do something simpler like what i suggested here
$\frac{d^3}{dx^3} x^3 = ?$
jolly parrotBOT
riemann
midnight haven
you can test out a few small numbers like n=3,4,5 and find a pattern
derive what
exactly
x^n
oh
midnight haven
**riemann**
well it's 1+2x^1 +3x^2 + 4x^3
so
sum from k to n of kx^(k-1)
ig
is the first derivative
of the sum
second will be k(k-1)x^(k-2)
so prob nth derivative will something like
n!x^(n-n)
but that's supposing that the nth derivative is the same n as the power
midnight haven
yeah x^n nth derivative will be n!
but
if you apply this to the sum earlier the x will disappear?
plush bramble
what does "earlier the x" mean
midnight haven
what does "earlier the x" mean
i just want to say i'm very sorry i feel like i'm giving you a hard time
midnight haven
.
midnight haven
ok
soooo
basically
first of all we had a function
we transformed it
into this sum using the formula for a geometric sequence
and we had to find the nth derivative of this sum
the derivative of a sum is the sum of the derivatives
so it's the nth derivative of x^2k
we found that the nth derivative of x^n = n!
let n=2k
the nth derivative of x^2k
will be 2k!
so it's the sum of (2k!)
but now since we don't have x anymore
we can't generally compute any derivative
using that formula
we only have the last one
rather than a general formula
this is my understanding so far
plush bramble
how do you know n is an even number
plush bramble
you can test out a few small numbers like n=3,4,5 and find a pattern
you really just need to actually write this out for a few terms like i said above
midnight haven
how do you know n is an even number
well because our original function was 1/(1-x^2) so the r=x^2 and will be the sum of (x^2)^k
i think
plush bramble
**riemann**
when you plug in r = x^2, find the 3rd, 4th, and 5th derivatives from here
plush bramble
well because our original function was 1/(1-x^2) so the r=x^2 and will be the su...
the power k or 2k after subbing in r= x^2 from geometric series has nothing to do with the n in the order of the derivative.
midnight haven
okkkkkkkkkkkkk
i think i finally understood
what you meant
bit late 😭 better later than never
let's check the pattern
i have to sleep i'll use this idea to solve it tomorrow i have classes at 8
i'm close
thanks
.close
pearl pondBOT
pearl pondBOT
Available help channel!
round pollen
Yeah so I did the first bit but no clue what the second bit is trying to do.
From the phrasing I'm inclined to believe the answer is no, but I built a (really bad) python program to check and it seems at least for some smaller c's it still works.
round pollen
this is what I did for the first part
also it's late and im kinda burnt out so like if I did something utterly stupid please dont bully me 🙏
pearl pondBOT
@round pollen Has your question been resolved?
pearl pondBOT
@round pollen Has your question been resolved?
cursive wraith
this is what I did for the first part
The justification for why one of x_a ± g lies in the set is not clear at all
cursive wraith
Say a = 4, b = 5. If I take g = 4 then neither of x_a ± g would be on the set if x_a is in the middle
Ofc I'm not allowed to take g = 4
Because g is a divisor of both a and b
You need to get a better upper bound for g
In order to obtain the result you want
round pollen
oh but you're saying x_a \pm g works though?
or are you not sure
I have no clue tbh this is so cooked
round pollen
oh well
cursive wraith
The problem is your justification for why it would
And as a hint, try to find a better upper bound for g
round pollen
probably if you can bound it above by b/2
cursive wraith
And can you?
round pollen
yeah cuz if you just recall that the gcd has prime factorisation where each prime's power is the min of the powers from a and b then since a< b at least one of a's powers must be smaller than the corresponding one for b and primes are at least 2
still have zero clue how to do second bit though
cursive wraith
still have zero clue how to do second bit though
If your intuition tells you it's true, try to prove it the same way you tried with the previous question
If you think it's false, come up with a counterexample
round pollen
i mean yeah no shit but like problem is I cant
I think i thsould be false
cuz the proof doesnt generalise so intuitively it feels so
and the wording feels sus
but i cant find a counterexample tbh
cursive wraith
but i cant find a counterexample tbh
So as a first hint, if you're looking for a counterexample, you should try to always look at the set {1,...,c}
Because if you offset by any value, you'll actually end up with "more factors"
And so more likely to be able to obtain a positive outcome
Whereas we want to find a counterexample, so an instance of a 'negative' outcome
cursive wraith
So as a first hint, if you're looking for a counterexample, you should try to al...
But for this set of consecutive integers you can take a,b,c
And their product is abc, a multiple of abc itself
Sorry I gtg now
round pollen
ok cya
cursive wraith
But for closing remarks, I suspect the claim is actually true
cursive wraith
And you should try starting with making a proof
round pollen
well a little bit but mostly because she asked me to help and i'm always too polite to say no so I always say yes
i mean it's fine because I don't do enough of this kind of stuff tbh
like number theoretic/ slightly competition ish flavoured thingies
cursive wraith
Mmmh
Ok so I just got new intel
The claim is false, but the first counterexample is pretty big
cursive wraith
you'll find it yourself but c = 143 is the one I found
round pollen
its like 7 nested for loops
round pollen
you'll find it yourself but c = 143 is the one I found
thats crazy
my computer will explode before that happens
the heat death of the universe will occur before my doodoo python program finds it
i do like 0 programming so its really bad
cursive wraith
Inefficient code can happen, it's fine
But yeah, for the partial solution:
a = 77, b = 91
I'll let you find the integer interval that works
cursive wraith
The first is in the thousands
Hope your code can at least handle it, given that a, b and c are fixed
round pollen
The first is in the thousand***s***
ggs
ok wait surely its >5000 right
cuz i just passed 5000
so like
i might have fucked somethinhg up 😭
i just hit 10k
round pollen
The first is in the thousand***s***
by thousands do you mean like 1k-10k or like 1000-999999
pearl pondBOT
@round pollen Has your question been resolved?
tiny condor
Let parallelogram ABCD exist. There is a point E, on line BC such that
BE:BC=1:2. Make line AE and BD. The cross point of line AE and BD shall be called point F. What is the ratio of the area the triangle FBE and Triangle AFB?
When I read the explanation, it says that "as EF:AF = 1:3, the ratio of the areas of the triangles △FBE and △AFB is 1:3". I know how EF:AF= 1:3, but idk how one makes that connection?
tiny condor
Here's the original question (s) if you care
quick star
> Let parallelogram ABCD exist. There is a point E, on line BC such that` BE:BC=...
well if that's the only info provided
isn't it the most natural thing to stumble upon?
u realize that the "height" is the same
so the area ratio is just dependent on the base ratio
tiny condor
AE is the base.. oohh. I was looking at it wrong, as if BE is the "base". Ty
.solved
pearl pondBOT
tiny condor
.closed
pearl pondBOT
Available help channel!
drowsy jay
@fair kindle
pearl pondBOT
<@1198704829703401474>
fair kindle
<@1198704829703401474>
Hello
drowsy jay
alright, so back to adding 5/6 and 2/3
remember, the LCM is the smallest number that 6 and 3 (the denominators) add up to
so, what is the LCM again?
fair kindle
so, what is the LCM again?
6
drowsy jay
6
alright!! one more time, among the denominators of 5/6 and 2/3, which denominator is lesser*?
drowsy jay
3
now you have the LCM 6, and the denominator 3
your goal now is to make 2/3 have the same denominator as 5/6
what number can you multiply to 3 to get 6?
fair kindle
what number can you multiply to 3 to get 6?
2
drowsy jay
2
okay okay, let's take a break and discuss another thing first
if I first multiply a number by 2, and then divide by 2, what happens to that number?
fair kindle
if I first multiply a number by 2, and then divide by 2, what happens to that nu...
Turns into 1
drowsy jay
Turns into 1
let me rephrase, for example, what do I get if I do (5 * 2) / 2?
5 is just an example number
dw, this is related. it might help you a lot
this is not a fraction by the way 😭
pearl pondBOT
@drowsy jay Has your question been resolved?
drowsy jay
brah
woven gorge
pearl pondBOT
woven gorge
help
toxic lichen
how do you normally find stationary pts
toxic lichen
ok and what is f'(x)
woven gorge
used pythag identity
toxic lichen
show don't tell
toxic lichen
yeah this is way overcomp lol
though hm i think you still end up with a cubic?
but also why not u := sin(x).
toxic lichen
you get 2u = 1/(1-u^2)
2u(1-u^2) = 1
you need to show somehow that this has no roots for 0<u<1
woven gorge
how?
toxic lichen
give it some thought
maybe do some more calculus on it somehow
i don't think it's gonna be good if i just give you the solution
woven gorge
maybe 2nd derivative test?
and how did u get thiss 2u = 1/(1-u^2)
@toxic lichen
can i see the solutoin
toxic lichen
u = sin(x) so cos^2(x)=1-u^2
ivory basin
<@&268886789983436800>
pearl pondBOT
ivory basin
Ah nvm too late
lavish hemlock
was that the crypto scam guy?
pearl pondBOT
Available help channel!
bitter herald
I am trying to compute this integral [
\int_0^t \6{\op{mod}}{\4{A\tau}{T_0}, A}\dd\tau
]
jolly parrotBOT
bitter herald
what should be the idea here
toxic lichen
are A and T_0 positive constants
so this is like... a sawtooth wave you're integrating?
bitter herald
so this is like... a sawtooth wave you're integrating?
yes
bitter herald
are A and T_0 positive constants
yes
so here is the generalised context
i am trying to plot [
\6\cos{2\pi f_c t + \int_0^t \6{\op{mod}}{\4{A\tau}{T_0}, A}\dd\tau}
]
jolly parrotBOT
bitter herald
for some frequency f_c
and a sawtooth defined by A and T_0
using desmos i get this but the integral (F) is discontinuous which doesnt make sense. so something must've gone wrong
i think my integral isnt right
toxic lichen
A*T0**^2**/2 floor(x/T0)?
wait hold up
uhh
no wait
2nd term i believe should be A/(2T_0) mod(x,T_0)^2
bitter herald
2nd term i believe should be A/(2T_0) mod(x,T_0)^2
oh you are right htanks
sharp smelt
Find $Aut(\Z_6)$
pearl pondBOT
Find $Aut(\Z_6)$
jolly parrotBOT
wai
sharp smelt
One obvious one is the identity homomorphism
I suppose I know have to ensure every element maps to an element of the same order
so 1 can only map to 1 or 5?
frozen bluff
yeah you're right
pearl pondBOT
crystal dew
<@&268886789983436800>
pearl pondBOT
Available help channel!
exotic gale
help please
pearl pondBOT
help please
ruby cargo
help please
!showyourwork
pearl pondBOT
help please
Show your work, and if possible, explain where you are stuck.
@exotic gale Has your question been resolved?
iron cobalt
@exotic gale you have any idea
compact ridge
whenever you see two terms being multiplied in the denominator, you want to make two different fractions with those terms
what technique allows you to do this?
inland ivy
pearl pondBOT
inland ivy
For part (a) is there an easier way other than calculating P(X + Y = n+1) and then P(X = k | X + Y = n+1) = p_X(k) p_y(n+1-k) / P(X+Y = n+1)?
I feel like the memoryless property of the geometric distribution will come into play here
pearl pondBOT
@inland ivy Has your question been resolved?
brittle tinsel
I don't think memorylessness will help here since your condition isn't an inequality of any kind
inland ivy
true
The thing is that it turns out to be 1/n, meaning it's a uniform distribution
so it really seems like there must be an easier way
brittle tinsel
oh that is interesting
inland ivy
yeah and I do have an explanation for it
If you expand out $\mathbb P(X+Y=n+1)$ as \begin{align*}
\mathbb P(X+Y=n+1) &= \sum_{k=1}^n p_X(k) p_Y(n+1-k) \
&= \sum_{k=1}^n p(1-p)^{k-1} p(1-p)^{n-k} \end{align*} the $k$'s cancel out, so every case is equally likely
jolly parrotBOT
kheer257
brittle tinsel
it's difficult to find an intuitive explanation, but it's basically saying that if you know it takes n+1 attempts to get 2 successes, getting the first success in any number of tries is equally likely
but it's not entirely clear to me why that would be the case
inland ivy
yeah
that's why I thought of memorylessness
but that doesn't seem to work
I guess I'll just ask my tutor
Also this one is just n * e^(-lambda) right?
brittle tinsel
yeah seems like it using binomial
fair kindle
dw, this is related. it might help you a lot
If I'm going to be honest Idk if you're in USA I don't understand how grades work. I'm Irish and I'm turning 13
meager scroll
Hey im starting to solve sequence equations current example is
a_n+2 - 4a_n+1 +4a_n = 3*2^n
a_0 =2 , a_1 = 5
I go on to solve the homogenous but the solution is r=2 for guess r^n
Then i try to make general form but i only have 1 solution from what i saw on the solution the general form is (A+Bn) 2^n
But why not A 2^n or better why Is that the general form
meager scroll
If someone can explain
feral olive
the particular solution should be of form a(2)^n
meager scroll
Are you sure?
feral olive
hang on
feral olive
so complementary solution
meager scroll
I do not know
I only recognise these terms from DEs
But i have no idea how they are assigned on sequence equations
feral olive
you make it into $\lambda^2-4\lambda+4=0$
jolly parrotBOT
ImOakley
meager scroll
I just asked chatgpt to find me a problem to see if i understand it
feral olive
$\lambda=2$
jolly parrotBOT
ImOakley
feral olive
here its basically just both solutions are 2
meager scroll
But with 1 solutioon idk
feral olive
wait sorry
meager scroll
here its basically just both solutions are 2
Yes ik
But then what is the general type
On ODEs
If its double root
You need to find some particular
Or wtv
feral olive
you change the form to $A(\lambda_1)^n+Bn(\lambda_2)^n$
jolly parrotBOT
ImOakley
meager scroll
Why though
feral olive
right lol
meager scroll
So (A+Bn) λ^n
feral olive
same thing though
meager scroll
This is what i sent
feral olive
anyway
meager scroll
But i dont understand why
feral olive
its something to keep it solvable
meager scroll
Can you explain
feral olive
it has to do with the linear algebra stuff
meager scroll
On ODEs
We assume maybe u(t) y1(t) is a solution
And we end up finding u(t) = t
So its Ae^r1t + B t e^r1 t
This looks much alike the (A+Bn)
Could it be something similar
feral olive
yeah ODEs are similar to difference equations, i did try learning the proof to how they work but im pretty rusty with it
meager scroll
So if i try maybe
u(n) r^n
Maybe
Lets see
No no
Try 2^n
The root i found
u(n) 2^n
I get u_n+2 - 2 u_n+1 +u_n =0
Idk if anyone can help prove that for repeated roots an(general) is (A+Bn) r^n pls help
Nvm i proved it
.close
pearl pondBOT
pearl pondBOT
Available help channel!
sharp smelt
pearl pondBOT
sharp smelt
the joint pdf would be $(e^{-x})(1-e^{-y})+ (e^{-y})(1-e^{-x})$
jolly parrotBOT
wai
pearl pondBOT
prisma prism
Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set:
a){80,83,86,89}
b){84,86,88,90}
c){79,81,83,85}
d){84,87,90,93}
fallen sundial
Out of all the patients in a hospital 89% are found to be suffering from heart a...
form a venn diagram
that should help
prisma prism
like man option b and option d set has 90% and 93% it is surely not possible
but the answer is c
fallen sundial
like man option b and option d set has 90% and 93% it is surely not possible
ye you can rule out b and d as it cant be larger than 89
fallen sundial
but the answer is c
prisma prism
ok
prisma prism
yeah the correct answer is c
prisma prism
how is it c then
that's why i posted this question
fallen sundial
prisma prism
the pic?
hybrid lance
but it can clearly not belong to b or d
It can belong to those
prisma prism
fallen sundial
It can belong to those
hmm like a single value
hybrid lance
It asks whether at least one of those values are possible, not necessarily all of them
fallen sundial
like one of the ones in set is true
prisma prism
It asks whether at least one of those values are possible, not necessarily all o...
then C should be possible too
fallen sundial
It asks whether at least one of those values are possible, not necessarily all o...
ic
hybrid lance
then C should be possible too
It's not
pastel depot
- im stuck on part c
- is the reason that part b is only 2 marks is because models are symmetrical and as U is the initial vertical velocity -U is it just before it hits the ground and same with horizontal being -3U (or is this not it)
agile kettle
like man option b and option d set has 90% and 93% it is surely not possible
try to eliminate thoese feelings when you practice!
oh oops.. i thought it was still that chanel
agile kettle
1. im stuck on part c
2. is the reason that part b is only 2 marks is because mo...
- is there something you have tried
- yeah pretty much you can directly answer b by knowing how projectile motion works under normal gravity conditions
its gonna have the same velocity ( so both the x y componenct just the y facing down now)
simple way to look at it is through energy conservation
initially the ball was on ground ( let the ground potential be M), it was given a speed of sqrt(U^2+ 3U^2) hence kinetic enegy
when its just falling on the ground its gonna have the same ground potential M , so it should also have the same kinetic energy hence the same speed but different velocity vector
pearl pondBOT
@pastel depot Has your question been resolved?
pastel depot
mb i had to go do something
pearl pondBOT
mb i had to go do something
pastel depot
2. yeah pretty much you can directly answer b by knowing how projectile motion w...
is this because it's symmetrical then
pastel depot
1. is there something you have tried
i mean i got sina and cosa but they don't help with anything for me as i don't form a quadratic using v=u+at
or can you work out S some other way
pearl pondBOT
@pastel depot Has your question been resolved?
pearl pondBOT
@pastel depot Has your question been resolved?
rough stream
If the ball is travelling 3m/s horizontally, how fast does it have to be moving vertically to get that angle?
Consider drawing a triangle
pastel depot
initial velocities are 14 and 42 horizontally and vertically
you can get what sina and cosa equal from tana = 1/4 but i dont see how that helps
you don't have S and it sounds like you should get a quadratic
or is this just not suvat at all
pearl pondBOT
@pastel depot Has your question been resolved?
pearl pondBOT
@pastel depot Has your question been resolved?
pearl pondBOT
@pastel depot Has your question been resolved?
prisma kernel
Well there's a formula for time taken
sacred osprey
pearl pondBOT
sacred osprey
i was told that the 2(x2-1) should multiply by 7 not 7y
i wonder why?
shouldn't they have the same dominator to work?
sacred osprey
what is the lcd (least common denominator) here?
i was told about lcd
but i dont remember what is that
jolly hill
i was told about lcd
the least common denominator (lcd) is the least common multiple (lcm) of the denominators, treating them as algebraic expressions here
so what is the lcm between 7y and 9y^2?
jolly hill
nope 63y isn't a multiple of 9y^2
jolly hill
do you have a video about lcd
basically 7y and 9y^2 need to be factors of whatever the least common multiple is
you can't multiply anything with 9y^2 to get 63y, you need an additional y
sacred osprey
so 7y x 9y^2?
pearl pondBOT
@sacred osprey Has your question been resolved?
sharp smelt
pearl pondBOT
sharp smelt
$\langle (T-\lambda I )(v),w \rangle= 0 \forall v,w$
\
$\iff \langle v, (T^* -\overline{ \lambda} I)(w) \rangle = 0 \forall v,w$
\
jolly parrotBOT
wai
sharp smelt
We then pick vectors ,v, that aren't orthogonal to $(T*-\overline{ \lambda} I)(w)$ and yet get 0.
tropic saddle
well v should be an eigenvector
jolly parrotBOT
wai
sharp smelt
well v should be an eigenvector
$T = \lambda I$ at v $\iff T^* = \overline{\lambda}I$ once more at v, so $T^*(v)= \overline{\lambda} v$
jolly parrotBOT
wai
sharp smelt
using this
tropic saddle
using this
but this holds as a whole map. why can you apply it in this way
sharp smelt
but this holds as a whole map. why can you apply it in this way
we can restrict the domain to span(v) that would still be a wel defined linear map
tropic saddle
but that's a different map now, so why should its adjoint have anything to do with T^*
sharp smelt
well, on the restricted domain, it would behave the same and be the same at every point
tropic saddle
show that
pearl pondBOT
sharp smelt
Thanks
pearl pondBOT
Available help channel!
shut flicker
im trying to show that 2^50 < 3^33, my first method, i tried showing that if 2^(5/3) is less than 3, then 2^50 < 3^33, but ig it doesn't work as my assumption that 2^(5/3) < 3 is wrong, i then tried using binomial expansion, but idk if this would be rigorous enough, although ig all terms (apart from 1) are bigger for the binomial expansion for 3^33 as we can distribute the 2s to the other terms, although i was wondering if there were more rigorous ways of showing this
pearl pondBOT
viscid shale
im trying to show that 2^50 < 3^33, my first method, i tried showing that if 2^(...
You can prob make use of log10
anyways. late
compact ridge
im trying to show that 2^50 < 3^33, my first method, i tried showing that if 2^(...
a nice trick is to try raising both sides to the same power
so 8^50 < 3^99 implies (8/9)^50 < 1/3
pearl pondBOT
Available help channel!
midnight haven
when a hint says "use the power series" what does it refer to? taylor? or what
compact ridge
when a hint says "use the power series" what does it refer to? taylor? or what
yes
assuming you're not in complex analysis or beyond
midnight haven
im not.'
well tough luck i dont know power series so i guess i will just pass..
thank you tho!!
compact ridge
thank you tho!!
no worries!
midnight haven
.close
wary vault
hello. if we had that for two vectors u and v:
uv = u^v + u•v , where ^ is the wedge product, then could we define the binomial theorem for (u+v)^n ? (yes its a dumb question)
tropic saddle
you mean like (u+v)(u+v)...(u+v) and each of those multiplications is the multiplication you wrote down?
tropic saddle
wait sry
I fucked up
well check whether uv satisfies all the rules you want for a multiplication
associativity, distributivity, commutativity
wary vault
the binomial theorem was defined even if we didnt have commutativity i think
its defined for rings or smth
tropic saddle
a theorem isnt "defined"
the usual statement that (a+b)^n = sum (n choose k) a^k b^(n-k) is wrong over non-commutative rings
wary vault
i mean our prof told us that we dont have to have commutativity when expanding (a+b)^n
tropic saddle
of course you can still state (a+b)^n = sum (all possible combinations) but at that point what are you even gaining
wary vault
youre right
tropic saddle
if the product you are defining is associative and distributive then sure feel free to expand out (u+v)(u+v)...(u+v)
no one is stopping you
pearl pondBOT
pearl pondBOT
Available help channel!
obtuse musk
i need help doing this mathscore, i dont know which point does it want to get the slope
obtuse musk
i mean it keep smaking my answer wrong i dont know which point does it wnat
lean oak
whichever two that are on the line.
obtuse musk
okay how about this one
obtuse musk
whichever two that are on the line.
yeah but it only gives 1 answer always
it doesnt accpet other points
cause it only wants that specific point of the answer
glass meadow
There's only one answer because it's asking for the slope, not for the points you use to get it
obtuse musk
which points do i use for the slope
glass meadow
whichever two that are on the line.
.
lean oak
yeah but it only gives 1 answer always
thing is though, your slope seems to be completely different from the answer. any two points on the same line must give the same slope.
(sorry, late reply, discord messed up my timing a little.)
lean oak
(-4, 0) is correct, but are you sure about (-2, 0)?
I don't see the line going through that point though.
not to mention, a line that goes through two points on the same y-coordinate must be a horizontal line, but this line isn't horizontal.
obtuse musk
(-4, 0) is correct, but are you sure about (-2, 0)?
yeah but look, it didnt use -4,0
the answer isnt correct purposely
lean oak
yeah but look, it didnt use -4,0
well, let me prove that then.
suppose I use (-4, 0) and (0, -2).
then, the slope is (-2 - 0)/(0 - (-4)) = (-2) / 4 = -1/2.
I've used (-4, 0), and I'm still getting the same slope as the answer given.
and that's because any two points lying on the same line will give you the slope of that line.
just because the explanation decides not to use (-4, 0) doesn't mean you can't, as long as you don't mess up the calculations.
if you gave a slope of -2, what was the second point you used to calculate the slope other than (-4, 0)?
lean oak
whichever two that are on the line.
which is what I said at the very start.
as long as your calculations are right and you didn't pick a point off the line, the slope will work out to be the same no matter which two points you pick.
lean oak
wdym?
obtuse musk
like these ones
lean oak
basically, points that the line passes through, yes.
real rose
for the following is my workong wrong
pearl pondBOT
for the following is my workong wrong
real rose
i did
an example answer is
i was wondering if theres any other way to attempt this besidees setting y= x^3 or = cx^3
please @ me
or ping:p
real rose
i did
like is this entirely wrong, afte r looking more i see that i suppose cuz numerator goes to 0 cuz r is there
so its always 0 anyways
for a and b the limits evaluet to 0 for context
pearl pondBOT
@real rose Has your question been resolved?
torpid wolf
this doesnt come with the answers
pearl pondBOT
this doesnt come with the answers
torpid wolf
so what i said is
to get a connected eulerian graph with orders 2, 2, 4, 4 a loop must be created which therefore implies that the graph is not simple
pearl pondBOT
@torpid wolf Has your question been resolved?
gloomy berry
Hi
crystal hill
hi, where are you facing trouble?
prisma kernel
What exactly do you need help with?
jolly hill
maclaurin series?
prisma kernel
Well, maybe someone else would be able to help you with that
gloomy berry
maclaurin series?
Yah
verbal whale
How to visualise this
Visualize what of them exactly?
gloomy berry
Taylor series
gloomy berry
Visualize what of them exactly?
These formulas
verbal whale
You mean how they approximate a function near a point?
gloomy berry
You mean how they approximate a function near a point?
Yes
verbal whale
Yes
Just go on Desmos or GeoGebra and plot the function together with the corresponding polynomial
jolly parrotBOT
∫_M dω = ∫_∂M ω
verbal whale
Just go on Desmos or GeoGebra and plot the function together with the correspond...
You can "stop" the poly at different degrees and you'll see that the higher the degree the better the approximation @gloomy berry
gloomy berry
Ok ty
verbal whale
Can this be valid
If you're approximating near 0 sure 👍
gloomy berry
Can we plot a graph
jolly hill
**∫\_M dω = ∫\_∂M ω**
and if you want to find the maclaurin series of any function, you can just use this
or for taylor uh
,tex $ \sum \limits_{n=0}^\infty \frac {{f^{(n)}} (a)}{n!} (x-a)^n $
gloomy berry
and if you want to find the maclaurin series of any function, you can just use t...
Yes it's formula for Taylor series
jolly parrotBOT
∫_M dω = ∫_∂M ω
gloomy berry
But sinx functions are included in it it not
verbal whale
Wdym?
gloomy berry
like trig functions
jolly hill
But sinx functions are included in it it not
a function must be smooth/infinitely differentiable at the expansion point, and its taylor series must converge to the actual function within some interval
gloomy berry
a function must be smooth/infinitely differentiable at the expansion point, and ...
Ok tyyy
verbal whale
They are mostly done on log and e
As you can see not at all, there are many other functions
gloomy berry
As you can see not at all, there are many other functions
Bruh I know they are done on trig but mostly on log
.close
pearl pondBOT
verbal whale
.reopen
pearl pondBOT
✅ Original question: #help-39 message
verbal whale
Bruh I know they are done on trig but mostly on log
Why mostly?
All of them are used with the same importance
verbal whale
But I have doubt in sinx
Which doubt?
gloomy berry
All of them are used with the same importance
Log is used as a function that is why so tremendously
sinx expansion
plotting
verbal whale
I don't really understand your doubt
gloomy berry
I don't really understand your doubt
The doubt is how to plot sinx Taylor series
Can you help
I understood the formula used when trig functions are there
@verbal whale
solid pier
The doubt is how to plot sinx Taylor series
what do you mean by "plot" here?
solid pier
But sinx functions are included in it it not
yes you can use this formula to find the coefficients for the taylor series for sin(x)
solid pier
the graph will be the same as the graph for sin(x)
pulsar flax
The taylor series converges to the function
verbal whale
sinx = x + y ....
Huh??
pearl pondBOT
solar coral
I know that show X, Y are independent <=> f_X(x)f_Y(y) = f_{X,Y}(x,y)
But I'm a bit confused on what I would do if Y>=0?
any guidance well appreciated
never mind i think i got it c:
.close
pearl pondBOT
pearl pondBOT
Available help channel!
royal harness
is this even solvable
pearl pondBOT
is this even solvable
verbal whale
Mmh those brackets are weird
toxic lichen
$\lim_{x \to 0^+} \frac{x \ln(x)}{x^x - 1}$
verbal whale
As it's written I'd really say no, it's not solvable
jolly parrotBOT
Ann
toxic lichen
im guessing this is what is meant?
toxic lichen
progress?
verbal whale
Hint: consider writing x^x as e^something
That way, you'll see how a substitution might help later
royal harness
Hint: consider writing x^x as e^something
e^ln(x^x) ?
verbal whale
Yes, but you can simplify that using one of the log rules
verbal whale
oh right e^xln(x)
Awesome 👍 (pay attention to brackets when writing in plain text though)
royal harness
Awesome 👍 (pay attention to brackets when writing in plain text though)
alr thanks!
@verbal whale sigh I can't seem to find the answer, I also tried to write the 1 as e^0 but 8 don't see what else I can do
I was absent during the lesson so that's also a big factor too ig
spare lark
What does it looks like
royal harness
this is what I get
.
serene sentinel
<@641337451826380809> sigh I can't seem to find the answer, I also tried to writ...
taylor series appx is a great way to go abt this I'd say
royal harness
taylor series appx is a great way to go abt this I'd say
we're not allowed to use Taylor series...
serene sentinel
ah I see
royal harness
we're allowed to use these identities tho
serene sentinel
we're allowed to use these identities tho
is definition of derivative ok?
royal harness
But with the e^ln
serene sentinel
and l'hopital perhaps?
royal harness
is definition of derivative ok?
no too
serene sentinel
hmm
royal harness
no one has been able to solve this one in our class
all I know is that this is equal to 1
using desmos
serene sentinel
well the trick I want to use is $e^y-1 = y$ for $y\to0$
jolly parrotBOT
Donkey
serene sentinel
yeah
that one trivializes the problem
but the issue is proving that identity without using taylor series or defn of derivative
royal harness
our teacher is sadistic what can I do sigh
and I didn't attend class cuz I was sick
serene sentinel
oh man
ok I think its provable using definition of e^y and $(1+x)^n \geq 1+nx$ for x>-1
jolly parrotBOT
Donkey
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
serene sentinel
$e^y=\lim_{n\to\infty}\left(1+\frac{y}{n}\right)^n$
jolly parrotBOT
Donkey
serene sentinel
$\left(1+\frac{y}{n}\right)^n \geq 1+n\frac{y}{n}=1+y$
jolly parrotBOT
Donkey
serene sentinel
so $e^y - 1\geq y \forall y\to0^+$
jolly parrotBOT
Donkey
serene sentinel
then apply $(1+x)^n \leq 1+ nx + \frac{n(n+1)}{2}x^2(1+x)^{n-2}$ and prove $e^y\leq1+y+\epsilon_n$ and $\epsilon_n\to0$
jolly parrotBOT
Donkey
serene sentinel
then you have $1+y \leq \left(1+\frac{y}{n}\right)^n \leq 1+y+\epsilon_n$
jolly parrotBOT
Donkey
serene sentinel
by squeeze theorem its proven that $\lim_{y\to0} \left[e^y-1=y\right]$
jolly parrotBOT
Donkey
peak mist
Hi I’m just wondering after I do a linear approximation using a known point, say x =6.5, for an unknown value, say x=6.3, and am determining whether it is an over approximation or under approximation, do I look at the concavity for 6.3 or 6.5 for the function?
plush bramble
!original
pearl pondBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
peak mist
I found L(6.3) = 47.56 using d=6.5
Since d = 6.3 falls within [5.75, 6.50] I know the function should be concave up at that point meaning it is an underestimation. But when I do that, do I look at d=6.5 or d=6.3?
Idk if that even makes sense
desert cliff
am I on the right trck for 19?
pearl pondBOT
am I on the right trck for 19?
dense jasper
pearl pondBOT
@desert cliff Has your question been resolved?
shell plover
could someone verify if the case 1 proof of fact has any version where it is valid?
with how it is written so far, i assume it is wrong since when we fix b = 0, then a = bn = 0 (aka a = 0), which cant be right since it goes against the given assumption a > 0
this question wants to prove the given fact with the basic property of divisibility (a = bn).
heavy onyx
yeah youre right, it isnt valid due to a>0
so your hypothesis would have to include b≠0
shell plover
damn alright dude, thank you
given whats shown in the other cases (b > 0 and b < 0), the given fact holds up
.close
pearl pondBOT